Product of Vectors Physics Notes

Product of Vectors:
A vector can be multiplied by a scalar. The components of the vector are multiplied by the scalar and the result is a scalar vector that is in the same direction as the original vector if the scalar is positive, or in the opposite direction if the scalar is negative.

A vector can also be multiplied by another vector. Two types of vector multiplication have been defined:

1. Scalar product
2. Vector product

1. Dot Product or Scalar Product of Two Vectors:
If the two vectors are inclined at the angle 0 then the dot product of two vectors is defined as the. product of their magnitude and cosine of the angle between them.
\(\vec{a}\) . \(\vec{b}\) =| \(\vec{a}\) || \(\vec{b}\) | cos θ …(i)

Where:
| \(\vec{a}\) | is the magnitude (length) of vector \(\vec{a}\)
| \(\vec{b}\) | is the magnitude (length) of vector \(\vec{b}\)
θ is the angle between \(\vec{a}\) and \(\vec{b}\)

So, we multiply the length of \(\vec{a}\) times the length of \(\vec{b}\), then multiply by the cosine of angle between \(\vec{a}\) and \(\vec{b}\). Since |\(\vec{a}\)|,|\(\vec{b}\)| and cos θ are scalars, so the dot product of \(\vec{a}\) and \(\vec{b}\) is a scalar quantity.

That is why dot product of two vectors is also called scalar product. Each vector, \(\vec{a}\) and \(\vec{b}\) has a direction, but their scalar product does not have a direction.

Why cos(θ)?
To multiply two vectors it makes sense to multiply their length together but only when they point in the same direction.

So we make one, “point in the same direction” as the other by multiplying by cos θ.

Scalar product of two vectors in term of their components:
∵ \(\vec{a}\). \(\vec{b}\) =| \(\vec{a}\) || \(\vec{b}\) | cos θ
= ab cosθ

= a(bcosθ) {∵ |\(\vec{a}\)| = a |\(\vec{b}\)| = b}
= (acosθ)b …(ii)
Therefore, \(\vec{a} \cdot \vec{b}\) = a(b cos θ) is the product of magnitude of \(\vec{a}\) and \(\vec{B}\) magnitude of component of \(\vec{B}\) along \(\vec{a}\) figure.

Alternating \(\vec{a} \cdot \vec{b}\) = b(acos0) is the product of magnitude of \(\vec{b}\) and the magnitude of component of \(\vec{a}\) along, \(\vec{b}\). Thus,

“Dot product of two vectors is also defined as the product of the magnitude of one vector and the magnitude of the component of other vector in the direction of first vector.”

Properties of dot product:
1. Commutative Property: From equations (i) and (ii), we have

2. Scalar Product of Perpendicular
Vectors: Let A and B be perpendicular to each other. So, the angle between them, 0 = 90°
∴ \(\vec{A} \cdot \vec{B}\) = ABcos 90°
or, \(\vec{A} \cdot \vec{B}\) = AB (0) (∵ cos 90° = 0)
⇒ \(\vec{A} \cdot \vec{B}\) = 0 …(ii)
So, when the unit vectors î, ĵ and k̂ are mutually perpendicular, then
î.ĵ = ĵ.k̂ = k̂.î = 0
and ĵ.î = k̂.ĵ = î.k̂ = 0 …(iii)

3. Scalar Product of Parallel Vectors: Let the vectors A and B be parallel to each other. Then, the angle between them, θ = 0°.
\(\vec{A} \cdot \vec{B}\) = AB cos0°
⇒ \(\vec{A} \cdot \vec{B}\) = AB (∵ cos0° = 1)
So, the scalar product of the two parallel vectors is equal to the product of the magnitudes of the two vectors.

4. Scalar Product of Equal Vectors: The scalar product of two equal vectors is equal to the square of the magnitude of any one of the vectors.
\(\vec{A} \cdot \vec{B}\) = AB cos0°
But, here A = B. So, A – B and θ = 0°
\(\vec{A} \cdot \vec{A}\) = AAcos 0° = A2
î.î = ĵ.ĵ = k̂.k̂ = 0………..(iv)

5. Scalar Product of Two Vectors In Terms of Their Components: The scalar product of two vectors:

6. Examples of Some Physical Quantities Obtained from the Scalar Product:
(a) The dot product of force and displacement is called the work.
W = \(\vec{F} \cdot \vec{S}\) where \(\vec{F}\) is the force and \(\vec{S}\) is the displacement vector.

(b) Power is the dot product of the force and the velocity.
p = \(\vec{F} \cdot \vec{v}\)

(c) The magnetic flux (Φ) associated with a plane is the dot product of the magnetic induction B and surface area A
ΦB = \(\vec{B} \cdot \vec{A}\)

(d) The electric flux (Φ) associated a plane is the dot product of thee electric field intensity (E) and the surface area (A).
∴ ΦE= \(\vec{E} \cdot \vec{A}\)

2. Vector Product or Cross Product of Two Vectors:
If we get a vector quantity on multiplying two vector quantities, then this product is called a vector product or a cross product. Vector product is expressed by putting a cross (X) mark between the two vectors.

The vector product of two vectors \(\vec{A}\) and \(\vec{B}\) is another vector C, whose magnitude is equal to the product of the magnitudes of the two vectors and sine of the angle between them.

Let the magnitudes of the vectors \(\vec{A}\) and \(\vec{B}\) be A and B, and the angle between them be θ. Then, the vector product is given as:
\(\vec{A} \times \vec{B}=|\vec{A}| \vec{B} \mid\) sin θ n̂
or \(\vec{A} \times \vec{B}=A B\) sin θ n̂ ………..(i)

Here, n̂ is the unit vector i\the direction of the magnitude of the resultant vector its direction is perpendicular to the plane of vector \(\vec{A}\) and \(\vec{B}\)

Physics Notes

Resolution of Vectors Physics Notes

Resolution of Vectors:
The process of vector addition by the triangle or parallelogram or polygon law is called composition of vectors.
The resolution of a vector into components vector is just the converse of the composition of vectors.
The process of spilliting up a vector into two or more vectors is known as resolution of a vector. The vectors into which a given vector is splitted are called the component vectors.

The component vectors in a given direction gives the measure of the effect of the vector in that direction. p
A given vector may be resolved into any number of component vectors. However we shall study the resolution of a vector into two and three-component vectors.

Resolution of vectors in Two

1. Resolution of a vector In two Dimensions: The resolution of a vector into two mutually perpendicular vectors is called the rectangular resolution of vector in a plane or two dimensions.

Consider that a vector \(\vec{OP}\)= \(\vec{A}\) has to be resolved into two component vector along the direction of two mutually perpendicular directions of X-axis and Y-axis. Let î and ĵ be the unit vectors along X-axis and Y-axis respectively figure

From point P, drop PM and PN perpendicular to X-axis and Y-axis respectively. From the parallelogram law of vector addition, it follows that

The equation (i) describes the rectangular resolution of the vector \(\vec{A}\) into the component vector A_x î and A_y ĵ. In practice Aî and Aĵ are called respectively x-component and y-component of vector A.

Further A_x, and A_y are called the magnitude of the two-component vectors.

If A is the magnitude of the vector \(\vec{A}\) and O is its inclination with X-axis, then from the right angled triangle OMP,

Adding the squares of Ax and Ay, we get

2. Resolution of a vector in Three Dimensions:
Consider that the vector \(\vec{OP}\)= \(\vec{A}\) represents a vector in space. In order to express it in the form of three mutually perpendicular components, construct a rectangular parallel OTBCDEP with three edges along the three co-ordinate axes OX, OY and OZ. Let î, ĵ, and k̂ be the unit vectors along OX, OY and OZ respectively as shown in the figure.

Then according to the polygon law of addition of vectors, we have
\(\overrightarrow{O P}=\overrightarrow{O T}+\overrightarrow{T B}+\overrightarrow{B P}\) ……(i)
If the three sides of the rectangular figure respectively are OT = A_x, OC = A_y and OE = A_z
Then, \(\vec{OT}\) = A_x î, \(\vec{OC}\) = A_y ĵ and \(\vec{OE}\) = A_z k̂

Since, \(\overrightarrow{T B}=\overrightarrow{O C}\) = A_y ĵ and
\(\overrightarrow{B P}=\overrightarrow{O E}=\overrightarrow{C D}\) = A_z k̂

Then the equation (i) becomes
\(\vec{A}\) = A_x î + A_y ĵ + A_z k̂ ……….(ii)

The equation (ii) expresses the vector \(\vec{A}\) oriented in space (three dimensions) in terms of its three rectangular components A_x î, A_y ĵ and A_z k̂.

Magnitude of \(\vec{A}\):
In triangle OBP
OP² = OB² + BP²

In triangle OCB
OB² = OC² + CB² = OC² + OT²
∴ OP² = OC² + OT² + BP²
A² = A_y² + A_x² + A_z²
A² = \(\sqrt{A_{x}^{2}+A_{y}^{2}+A_{z}^{2}}\) …….(iii)
Thus the magnitude of a vector is equal to the square root of the sum of the squares of the magnitude of its rectangular components.

Directions: Cosines of a Vector:
If oc,P and y are the angles which \(\vec{A}\) makes with X, Y and Z axes respectively, then
cos α = \(\frac{A_{x}}{A}\) or A_x = A cos α
cos β = \(\frac{A_{y}}{A}\) or A_y = A cos β
cos γ = \(\frac{A_{z}}{A}\) or A_z = A cos γ

Here cos α, cos β and cos γ are called the direction cosines of the vector \(\vec{A}\)
Putting the value of A_x, A_y and A_z in eq. (iii), we get,
A² = A² cos² α + A² cos² β + A² cos² γ
A² = A² (cos² α + cos² β + cos² γ )
It means that the sum of the squares of the direction cosines of a vector is always unity (1).

Physics Notes

AnalytIcal (Mathematical) Method of Vector Addition Physics Notes

Analytical (Mathematical) Method of Vector Addition:
1. Triangle Law of Vector Addition: Let us consider the two vectors \(\vec{P}\) and \(\vec{Q}\), inclined at angle θ, be acting on a particle at the same time. Let them be represented in magnitude and direction by two sides

\(\vec{OA}\) and \(\vec{AB}\) of triangle OAB, taken in the same order, figure. Then according to the triangle law of vectors addition, the resultant \(\vec{R}\) is represented by the side \(\vec{OB}\) of the triangle, taken in opposite order with the omagnitude and direction.

Addition of vectors

To find out the magnitude of the resultant:
From the point B draw BN perpendicular to OA, meeting at point N, when produced. Let ∠BAN = 0 then, from the right angled triangle ONB, we have
OB² = (ON)² + (NB)²
= (OA + AN)² + (NB)²
(OB)² = (OA)² + (AN)² + 2OA AN + (NB)²
Now, (AN)²+ (NB)² = (AB)²
⇒ (OB)² = (OA)² + (AN)² + 2(OA)(AN) + (NB)² …(i)
Using the Pythagoras theorem in right angled Δ ANB,
(AB)² = (AN)² + (NB)² …(ii)
But, cos θ = \(\frac{A N}{A B}\)
AN = AB cos θ …(iii)

From equations (i), (ii) and (iii) we get
(OB)² = (OA)² + 2 (OA)(ABcosθ) + (AB)²
or, R² = P² + 2PQcosθ + Q²
or, R² = P² + Q² + 2PQcosθ
or, R = P² + Q² + 2PQcosθ …(iv)

The above expression is known as the law of cosines.

To determine the direction of the resultant vector
From the right-angled Δ ONB
tan α = \(\frac{N B}{O N}\)
or, tan α = \(\frac{N B}{O A+A N}\) ……(v)

Special Cases:
1. When \(\vec{P}\) and \(\vec{Q}\) are in the same direction, then θ = 0°.
From equation (iv), we have

2. When P and are perpendicular to each other, then θ = 90°,
From equation (iv), we have

3. When \(\vec{p}\) and \(\vec{Q}\) are in opposite directions. Then,
θ = 180°,
From equation (iv),

⇒ R = \(\)
or, R = P – Q………(xiii)
Here, the value of R is minimum.
From equation (viii), we get
α = tan^-1 \(\left(\frac{Q \sin 180^{\circ}}{P+Q \cos 180^{\circ}}\right)\)
or, α = 0° when P > Q
and, α = 180° when P < Q ….(xiv)
It may be pointed out that the magnitude of the resultant of two vectors is maximum, when they act along the same direction and minimum, when they act along the opposite direction.

2. Parallelogram Law:
Proof: The proof for the resultant vector in parallelogram addition is as follows:
Let the two vectors \(\vec{P}\) & \(\vec{Q}\) inclined to each other at an angle 0 be represented in magnitude as well as in the direction by the sides \(\vec{OA}\) and \(\vec{OC}\) of the parallelogram OABC [figure], Then according to the parallelogram law, the resultant of \(\vec{P}\) and \(\vec{Q}\) is represented both in magnitude and direction by the diagonal \(\vec{OB}\) of the parallelogram.

To find out the magnitude: From point B draw a perpendicular BN on OA, meeting OA at point N.
From the right-angled Δ ONB
OB² = ON² + NB²
But ON = OA + AN
From Δ ANB


It is known as the law of cosines.

The direction of the Resultant:
Suppose that the resultant R makes an angle a with the direction of vector P. Then from the right-angled triangle ONB
tan α = \(\frac{B N}{O N}=\frac{B N}{O A+A N}=\frac{Q \sin \theta}{P+Q \cos \theta}\)

1. It should also be noted that while finding the resultant of two vectors by the parallelogram law, the two vectors \(\vec{P}\) and \(\vec{Q}\) have to be co-initial vectors.

2. The magnitude of the resultant of two vectors is maximum, when angle between them is 00 and is minimum when they act is oppsiste directions
i.e., θ=180° 0
Rmax = P + Q
Rmin = P – Q

3. The resultant of two equal vectors can be zero if they act in opposite direction.

4. The number of three unequal vectors can be zero. If they are coplanar and they can be represented by three sides of a triangle taken in one order.

5. If \(\vec{P}+\vec{Q}=\vec{P}-\vec{Q}\), then vector \(\vec{Q}\) must be zero vector.

6. If \(|\vec{P}+\vec{Q}|=|\vec{P}-\vec{Q}|\) then vectors \(\vec{P}\) and \(\vec{Q}\) must be at right angle to each other.

Physics Notes

Combination of Vectors Physics Notes

Graphical Method:
The following facts should be followed while combining the vectors:

1. When a vector is displaced at any point within the parallel space without changing its magnitude then, the vector remains unchanged.
2. Addition of two vectors always give a vector. It is called the resultant vector.
3. Only those vectors that represent similar physical quantities can be added.

The laws of addition of vectors are:
(a) Triangle Law of Vector Addition
(b) Parallelogram law of Vector Addition
(c) Polygon law of Vector Addition

(a) Triangle Law of Vector Addition:
It states that, “If two vectors acting on a body are represented both in magnitude and direction by two sides of a triangle taken out in an same order then the resultant is represented by the third side of the triangle taken in the opposite order in magnitude and direction.”

This simply means that, if you have two vectors that represents the two sides of the triangle then the third side of that triangle will represent their resultant.

Figure (a) shows the two vectors \(\vec{P}\) and \(\vec{Q}\). To find out their resultant (sum) by using the triangle law of vector addition, draw the vector \(\vec{AB}\) = \(\vec{P}\). Now move the vector \(\vec{Q}\) parallel to itself, such that its tail coincides with the tip of \(\vec{P}\) vector. Mark the tip of vector \(\vec{Q}\) as C [figure (b)]. Thus the vector \(\vec{P}\) and \(\vec{Q}\) have been represented by the two sides \(\vec{AB}\) and \(\vec{BC}\) or a triangle taken is the same order.

Therefore according to the law of vector addition, the vector \(\vec{AC}\) = \(\vec{R}\) (say) drawn from the tail of the vector P to the tip of the vector Q is the third side of the triangle taken in the opposite order and hence it represents the sum or the resultant of the vectors \(\vec{P}\) and Q.
Thus, \(\vec{P}+\vec{Q}=\vec{R}\)

(b) Parallelogram law of Vector Addition:
The parallelogram law is a slightly more explanation of the triangular law.

It states that if “the two vectors are considered to be the adjacent sides of a parallelogram, then, the resultant of the two vectors is given by the vector which is a diagonal passing through the point of contact of the two vectors.”

Let us find out the resultant of the two vectors \(\vec{P}\) and \(\vec{Q}\) as shown in figure (a). To find out the resultant, draw the vector \(\vec{AB}\) = \(\vec{P}\). Now move the vector \(\vec{Q}\) parallel to itself so that its tail coincides with the tail of vector \(\vec{P}\). If we mark its tip as D, then the vector AD represents the vector \(\vec{Q}\). Complete the parallelogram ABCD as shown in figure (b)

Thus, the vector \(\vec{P}\) and \(\vec{Q}\) have been represented by the two adjacent sides \(\vec{AB}\) and \(\vec{AD}\) respectively of a parallelogram. Therefore, according to the parallelogram law of vector addition, the vector \(\vec{AC}\) = \(\vec{R}\) (say), which represents the diagonal of the parallelogram passing through the point A, represents the sum or the resultant of the vector \(\vec{P}\) and \(\vec{Q}\)

Thus \(\vec{P}+\vec{Q}=\vec{R}\)
Note: According to the parallelogram law of addition of vectors \(\overrightarrow{A B}+\overrightarrow{A D}=\overrightarrow{A C}\)

Since, in a parallelogram, the opposite sides are equal and parallel, they must represent equal vectors
∴ \(\overrightarrow{A D}=\overrightarrow{B C}\)
In the above equation, setting \(\overrightarrow{A D}=\overrightarrow{B C}\), we have
\(\overrightarrow{A B}+\overrightarrow{B C}=\overrightarrow{A C}\)

It is the triangle law of vector addition. Hence triangle law of vector addition follows from the parallglogram law of vector addition.

(c) Polygon Law of Vector Addition:
The polygon law is an extension of the earlier two laws of vector addition. It is successive application of the triangle law to more than two vectors. It states that, “If a number of vectors can be represented in magnitude and direction by the sides of a polygon taken out in the same order. Then their resultant is represented in magnitude and direction by the closing side of the polygon taken in the opposite order.”

Definition 2: Polygon law: If (n-1) sides of a polygon is in sequence. Then the nth side, closing the polygon in the opposite direction, represents the sum of the vectors in both magnitude and direction.

Polygon Law

Let us find out the resultant of the four vectors namely \(\vec{a}, \vec{b}, \vec{c}\) and \(\vec{d}\) as shown m the figure (A). To fmd out their resultant, draw the vector \(\vec{OP}\) = \(\vec{a}\). Now, move the vector \(\vec{b}\) parallel to itself, so that its tail comcides with the tip of vector \(\vec{a}\). Mark the tip of vector \(\vec{b}\) as Q. Now move the vector \(\vec{c}\) parallel to itself, so that its tail coincides with the tip of the vector \(\vec{b}\) and mark the tip of vector \(\vec{c}\) as S finally move the vector \(\vec{d}\) parallel to itself, so that the tail of vector \(\vec{d}\) coincides with the tip of vector \(\vec{c}\). Then the vectors \(\vec{a}, \vec{b}, \vec{c}\) and \(\vec{d}\) have been represented by the sides \(\overrightarrow{O P}, \overrightarrow{P Q}, \overrightarrow{Q S}\) and \(\vec{ST}\) taken in the same order figure (b)

According to the polygon law of vector addition, the closing side \(\vec{OT}\) = \(\vec{R}\) (say) is taken in opposite order represents the sum or the resultant of vector \(\vec{a}, \vec{b}, \vec{c}\) and \(\vec{d}\)
i.e. \(\vec{a}+\vec{b}+\vec{c}+\vec{d}=\vec{R}\)
Proof: In triangle OPQ, the vectors a and b have been represented by the sides OP and PQ taken in the same order. Therefore, from the triangle law of vector addition, the closing side OQ is taken in the opposite order represents the resultant of vectors a and b. Thus,

Physics Notes

One Dimensional, Two Dimensional and Three Dimensional Vectors in Cartesian Coordinate System Physics Notes

One Dimensional, Two Dimensional and Three Dimensional Vectors in Cartesian Coordinate System:
A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a pair by a pair of coordinates, which are the distances to the point from two fixed perpendicular directed lines. Each reference line is called the axis of the system, and the point where they meet is called its origin (0, 0).

Cartesian Coordinate System

1. One Dimensional Vectors: A vector that is directed along one of the axes (X-axis or Y-axis or Z-axis), is called a one-dimensional vector.

• Along X-axis, \(\vec{r}\) = xî,
• along Y-axis, \(\vec{r}\) = y ĵ, and
• along Z-axis, \(\vec{r}\)= z k̂.

One Dimensional Vectors

2. Two Dimensional Vectors: The vectors that are situated in a plane are called two-dimensional vectors. Two dimensional vectors are directed in two directions or the two axis. If the vector \(\vec{r}\) is located in X-Y plane, then the two dimensional vector \(\vec{r}\) is represented as \(\vec{r}\) = xî + yĵ, if \(\vec{r}\) is in Y-Z plane, then \(\vec{r}\) = yĵ + zk̂ and in X-Z plane, \(\vec{r}\) = xî + zk̂.

• X-Y plane, \(\vec{r}\) = xî + yĵ
• In Y-Z plane, \(\vec{r}\) =yĵ + zk̂
• In X-Z plane, \(\vec{r}\) = xî + zk̂

Two Dimensional Vectors

3. Three Dimensional Vectors: A three dimensional vector is directed in all the three axis (X-axis, Y-axis and Z-axis). If the location of point P(x, y, z) is to be represented by the vector \(\vec{r}\). Then, \(\vec{r}\) = xî + yĵ + zk̂ and hence, \(\vec{r}\) is a three-dimensional vector. Here, x, y and z are the coordinates that have the numerical values.

Three Dimensional Vectors

Physics Notes

Representation of Vectors Physics Notes

Representation of Vectors:
A vector can be represented by a straight line with arrow head on it, i.e., an arrowed line. Here the length of line drawn on suitable scale represents the magnitude of vector and head of arrow represnets the direction of a vector.
If the vector represnts a directed distance or displacement from point A to a point B (see in figure below)
It can also be denoted as \(\overrightarrow{\mathrm{A}}\)B or \(\overrightarrow{a}\).

Here the point A is called the origin, tail, base or initial point. Point B is called the head, tip, end point, terminal point or final point. The length of the arrow is proportional to the vectors magnitude, while the direction in which the arrow points indicates the vector’s direction.

To represent a physical quantity in a graph, we need to have a suitable scale for it. For example, to represent displacement of 20 m in East direction, it is not possible to draw a straight line of 20 m on a sheet. To represent this vector quantity, we will have to take a suitable scale, we can take displacement of 10 m = 1 cm.

So, 20 m of displacement (vector quantity) can be shown by drawing a line of 2 cm moving from west to east direction. Similarly, if a force of 6 N is applied on an object in north direction, and we take 3 N = 1 cm, then it can be shown on a graph by drawing a line of 2 cm, moving from south to north direction.

Some Important Definitions Related with Vectors
1. Modulus of a vector: The magnitude of vector is called modulus of a vector. The modulus of a vector \(\overrightarrow{A B}\) is represented by |\(\overrightarrow{A B}\)| (putting vertical lines on both sides of the vectors) or |AB|.

2. Negative vector: The negative of a vector is defined as another vector having the same length (magnitude) but drawn in the opposite direction.

The negative of a vector \(\overrightarrow{A B}\) is represented as – \(\overrightarrow{A B}\). The two vectors have equal magnitude but their directions are opposite. If the vector \(\overrightarrow{A B}\) is from west to east, then the vector – \(\overrightarrow{A B}\) is from east to west. The angle between the negative vectors is n rad or 180°.

3. Equal vectors: Two y, vectors are said to be equal, if they have the same magnitude and the same direction.
Figure shows the two vectors \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathrm{B}}\) having the o same magnitude and the same direction, therefore

\(\vec{A}=\vec{B}\) the vectors \(\vec{A}\) and \(\vec{B}\) are equal vectors.

Note:

• The angle between equal vectors is 0°.
• If a vector is moved parallel to itself, it represents a vector equal to itself i.e. the same vector.
• For two vectors to be equal, it does not matter, whether the two vectors have their tails at the same point or not. If the scale selected for both1 the vectors is the same they are represented by two equal and parallel lines with arrowheads in the same direction.

4. Unequal vectors: Two vectors are said to be unequal vectors. If they have equal magnitude but the directions are opposite or they have equal direction but magnitudes are different or they have unequal magnitudes and directions.

5. Unit vector: Vector quantities have direction and magnitude. However, sometimes one is interested only in the direction of the vector and not in the magnitdue. In such cases, for convenience, vectors are often “normalized” to be of unit length.

Every non-zero vector has a corresponding unit vector, which has the same direction as that vector but a magnitude of 1 (unit).

A unit vector of a is written as a and is read as \(\vec{a}\) cap’ or “a hat”. Since magnitude of \(\vec{a}\) is a, hence.

• The magnitude of a unit vector is always 1.
• A unit vector of a given vector tells the direction of that vector.
• A unit vector has no units and no dimensions.

In cartesian coordinates, î, ĵ, k̂ are the unit vectors along the x-axis, y-axis, and z-axis respectively.

• î = a unit vector directed along the positive X-axis
• ĵ = a unit vector directed along the positive Y-axis
• k̂ = a unit vector directed along the positive Z-axis

These unit vectors are commonly used to indicate directions, with a scalar coefficient providing the magnitude.

Method to calculate the unit vector of a vector: Consider a point P in the coordinate system.
Magnitude the position vector (OP)
\(\vec{r}\) = xî + yĵ + zk̂

Magnitude = distance between the points O and P

6. Co-initial vectors: Two vectors are said to be co-initial if they have a common initial point. In figure the two vectors A and B have been drawn from the same point ‘O’. They are called co-initial vectors.

7. Collinear vectors: Two vectors having equal or unequal magnitudes, which either act along the same line [figure (a)] or along the parallel line in the same direction [figure (b)] or along the parallel lines in the opposite direction [figure (c)] are called collinear vectors.

8. Zero vector or null vector: A vector with zero magnitude is called a null vector or a zero vector. The direction of a null vector is undefined. It can be along any direction.

A zero vector is represented by \(\vec{0}\) (arrow over the number zero) .

We know that a vector may be representedby an arrow, the length of the arrow representing its magnitude and the arrowhead representing the direction of the vector. Since the zero vector has no magnitude, the arrow representing the zero vector has to be of zero length. As such, the direction of the arrow head can not be specified. Therefore, the zero vector has zero magnitudes and may be said to have an arbitrary direction.

The following two operations give rise to a zero vector:
1. When the negative of a vector is added to the vector, the result is a zero vector,
Thus \(\vec{P}\) + (- \(\vec{P}\)) = 0

In vector space, the additive inverse does exist. If there is a vector \(\vec{P}\) and a vector – \(\vec{P}\) then there must exist zero vector such that
\(\vec{P}\) + (- \(\vec{P}\)) = \(\vec{0}\)

2. When a vector is multiplied by zero, the result is a zero vector. Thus,
0(\(\vec{P}\)) = \(\vec{0}\)

Examples of Zero Vector

• Two people pulling a rope in the opposite directions with equal force.
• Displacement of throwing an object upward and then again holding it at the same position.
• The velocity of a train standing still on a platform.
• Acceleration of a car going at a uniform speed.
• The position vector of the origin of Coordinate axis is a zero vector.

Physics Notes

Scalar and Vector Quantities Physics Notes

Scalar Quantities:
The physical quantities, which have magnitude but no direction are called scalars, Mass, length, distance covered, time, density, work, temperature, charge, specific heat, energy, power, speed, length etc. are few examples of scalars, A scalar quantity can be completely described by a number; representing its magnitude. A scalar may be positive or negative. They can be added, subtracted, multiplied and divided according to the simple rules of Algebra.

Vector Quantities:
The Physical quantities which have both magnitude and direction are called vectors. Displacement, velocity, acceleration, force, momentum, electric field, impulse, gravitational field etc. are few exmaples of vector quantities. It may be pointed out that the vectors can not be added, subtracted, multiplied or divided as one may do in case of scalars. It is because, in addition to magnitude, vectors have direction also. Vectors are added, subtracted and multiplied according to the rules of Vector Algebra. The division of a vector by another vector is not a valid operation m Vector Algebra.

1. A physical quantity possesing both magnitude and direction is not necessarily a vector. For example, electric current and pressure. This is; because vectors obey the rules of Vector Algebra and it is possible that a physical quantity which has both mangitude and direction may not obey the vector laws.

2. There are some physical quantities which are neither a scalar nor a vector. These are called Tensor quantities. It does not have one specified direction but have different values in different directions. Examples: Moment of inertia, elasticity constant, stress etc.

Vectors can be divided into two parts:

1. Polar vectors
2. Axial vectors

1. Polar vectors: Polar vectors are those vectors which have a fixed starting point or a point of application and act along the direction of motion of the body.
For example: Displacement, force etc. are polar vectors.

(a) Polar Vectors

2. Axial vectors: Axial vectors are those vectors which represent rotational effect and act along the axis of rotation in accordance with the “right-hand screw rule”.
For example Angular velocity, angular acceleration, torque, angular momentum, etc. are axial vectors.

(b)
A vector having an anticlockwise rotational ‘ effect, will have its direction along the axis of rotation as shown in figure (b)

Note: Axial vectors describe rotational motion and act along the axis of rotation (according to right-hand screw rule).

Polar vectors describe translatory motion and have fixed starting point. The direction of polar vector remain unchanged irrespective of the coordinate system choosen.

Physics Notes

Basic Mathematical Concepts Notes

Physics deals with the study of nature. It tells about the fundamental laws of nature that govern the various physical phenomena in the world. Measurement of such a physical quantity involves the comparison with a basic and internationally accepted reference standard called UNIT. A physical quantity is expressed by a number (or numerical measure) accompanied by a unit.

For example, if the mass of an object is 7 kg. Then, it means that the mass of the object is 7 times the unit (i.e., 7 times of 1 kg).

But, there are some physical quantities that can not be completely expressed by the number and the unit. For example, let us consider that a boy Abhimanyu moves from a reference point O to cover a distance of 100 m. The distance between the point O to 100 m can be in infinite directions. The figure shows some of the displacements by Abhimanyu as P₁, P₂, P₃,…, etc.

It is apparent that without knowing about the direction of motion of Abhimanyu, we can not say anything about the final position of Abhimanyu.

All the measurable physical quantities can be divided into two classes, namely: i. Scalar quantities 2. Vector quantities

→ Scalar Quantities: Scalars are quantities that are fully described by a magnitude (or numerical value) only.
Example: Distance, mass, density, etc.

→ Vector quantities: The physical quantities which require both magnitude and direction for their description are called vector quantities.

→ Kinds of vectors:

• Axial vectors: Axial vectors are used to describe rotational motion. These are those which represent rotational effect and act along the axis of rotation according to the right hand screw rule example: angular velocity (\(\overrightarrow{(\omega)}\)), angular acceleration (\(\overrightarrow{(α)}\)), etc.
• Polar vectors: Polar vectors describe translation motion and have a starting point. The direction of the polar vector remains unchanged irrespective of the coordinate system chosen. Example: displacement, force, etc.

→ Some important definitions regarding vectors:

1. Equal vector: Two vectors are said to be equal if they have the same magnitude and direction.
2. Opposition negative vector: A negative or opposite vector is defined as a vector having the same length but drawn in opposite direction.
3. Unequal vector: Two vectors are said to be unequal vectors; if they have equal magnitude but opposite direction or they have equal direction but opposite magnitudes or they have unequal magnitudes and direction.
4. Zero vector: A vector whose magnitude is zero is called a zero vector.
5. Unit vector: A vector that has a magnitude of one.

→ Addition of vectors

1. Triangle law of vector addition: It states that when two vectors are represented by two sides of a triangle in magnitude and direction taken in the same order then the third side of that triangle represents in magnitude and direction the resultant of the vectors.
2. Parallelogram law of vector addition: It states that if two vectors are considered to be the adjacent sides of a parallelogram, then the resultant of two vectors is given by the vector which is a diagonal passing through the point of contact of two vectors.
3. Polygon law of vector addition: This law is used to add more than two vectors. It states that if two or more vectors are represented by the adjacent sides of a polygon, taken in the same order both in magnitude and direction, then the result is given by the closing side of the polygon taken in opposite order both in magnitude and direction.

→ Subtraction of vectors: The process of subtracting one vector from the other is equivalent to adding, vectorially, the negative of the vector to be subtracted.

→ Product of vectors:

• Scalar product or dot product: The scalar product or the dot product is the product of two vectors which are given magnitude only. The dot product is written using a central dot (.).
  \(\overrightarrow{\mathrm{A}}\). \(\overrightarrow{\mathrm{B}}\) =|\(\overrightarrow{\mathrm{A}}\)||\(\overrightarrow{\mathrm{B}}\)| cosθ = ABcosθ
• Cross product or vector product: The vector product or cross product is the product of two vectors which give a physical quantity which has both magnitude and direction.
  or \(\overrightarrow{\mathrm{A}}\) x \(\overrightarrow{\mathrm{B}}\) = AB sinθ.n̂

→ Differential calculus: It is a subfield of calculus concerned with the study of the rates at which quantities change.

→ Integral calculus: A branch of mathematics concerned with the theory and applications of integrals and integration.

→ Scalar Quantity: Those quantities which required only magnitude for their description are called scalar quantities.

→ Vector quantity: Those physical quantities which are described both by direction and magnitude are called vector quantities.

→ Tensor quantity: Those quantities which are note even clearly defined by magnitude and direction are called tensor quantities.

→ Axial vectors: A quantity that transforms like a vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection.

→ Polar vectors: Those vectors whose initial point is definite are “ailed polar vectors.

→ Zero vector: Those vectors whose magnitude is zero.

→ Unit vector: Those vectors whose magnitude is one.

→ Position vector: Vector representing the position or point or any object is called the position vector.

Physics Notes

Significant Figures and Rounding off The Digits Physics Notes

Significant Figures:
The significant figures express the accuracy with which the physical quantity may be expressed. They are the digits which give us useful information about the accuracy of measurement.

The greater the number of significant figures obtained when making a measurement, more accurate is the measurement, conversely, a measurement made to only few significant figure is not a very accurate one. For example, a recorded figure of 5.32 means the quantity can be relied on as accurate to three significant figures and a figure of 5.321 is said to be accurate to four significant number.

The following rules have been setup for determining the number of significant figures :
1. All non-zero digits are significant. 243.48 contains five significant figures.

2. All zeros occuring between two non-zero digits are significant.

3. All zeros to the right of a decimal point and to the left of a non-zero digit are never significant 0.00678 contains three significant figures. The single zero conventionally placed to the left of the decimal point in such an expression is also never significant.

4. (a) All zeros to the right of a decimal point are significant if they are not followed by a non-zero digit. For example 30.00 contains four significant figures.
(b) All zeros to the right of the last non-zero digit after the decimal point are significant. For Example 0.054300 contains five significant figures.

5. (a) All zeros to the right of the last (rightmost) non-zero digit are not significant. Ex. 3030 contains three significant figures.
(b) All zeros to the right of the last non-zero digit are significant, if they come from a measurement.

Suppose that the distance between two objects is measured to be 3030 m. Then 3030 m contains four significant figures.

Change of units does not change the number of significant figures in a measurement.
For example, the length x = 2.308 cm has four siginificant digits. In different units, the same length can be written as x = 23.08 mm: x = 0.00002308 km. All these numbers have the same number of significant figures namely four, the digits 2, 3, 0 and 8.

Significant figures in addition and subtraction:
The accuracy of a sum or a difference is limited to the accuracy of the least accurate observation in the addition and subtraction.

Rule: Do not retain a greater number of decimal places in a result computed from addition and subtraction than in observation; which has the fewest decimal places.

Illustration: Add and substract 428.5 and 17.23 with due regards to significant figures.
We have 428.5 + 17.23 = 445.73 and 428.5-17.23 = 411.27
But in physics, the sum and difference taken in this manner are discouraged. In fact, in the data 428.5, we have assumed zero to be in second place after decimal. The data 428.5 might have been written to first decimal only because of the inability of the instrument to measure it to the further accuracy. Therefore the choice of zero only in the second decimal place of data is not justified.

To add or subtract data in such a situation there are two methods:
1. By rounding off the answer: The data 428.5 is the weakest link as its value is known upto first decimal only. Therefore the answer should also be retained only up to first decimal place.
Sum = 428.5 +17.23 ⇒ 445.73
Difference = 428.5 -17.23 ⇒ 411.27

As said earlier, in case the second decimal is occupied by 5 or more than 5. The number in first decimal is increased by 1. On the other hand if the second decimal is occupied by a number less than 5. it is ignored.

Rounding off the result of the above sum and difference to first decimal, we have correct sum = 445.7 and correct difference = 411.3

2. By rounding off the other data: The result can also be obtained by rounding off the other data in accordance with the data. Which is the weakest link. The data 17.23 should be rounded off to 17.2 (3 in second decimal place is ignored) and the added to or substracted from 428.5. Thus we have:
correct sum ⇒ 428.5 + 17.2 ⇒ 445.7 and
correct difference ⇒ 428.5 -17.2 ⇒ 411.3

Significant figures in multiplication and division: The following rule applies for multiplication and division:

Rule: The least number of significant figure in any number of the problem determines the number of significant figures in the answers. This means you must know that to recognise significant figures in order to use this rule.

Rounding Off the Digits:
Rounding off a number is done to obtain its value with a definite number of significant figures. For this following are the rules:

1. If the digit to drop is less than 5, then the preceding digit is not changed.
   For e.g., 1.24 is rounded off to 1.2.
2. If the digit to drop is greater than 5, than the preceding digit is raised by 1.
   For e.g., 19.48 is rounded off to 19.5.
3. If the digit to drop is 5 and the preceding digit is even, then it is not changed.
   For e.g., 1.25 is rounded off to 1.2.
4. If the digit to drop is 5 and the preceding digit is odd, then it is increased by 1.
   For e.g., 3.35 is rounded off to 3.4.

Physics Notes

Absolute Error, Relative Error, Percentage Error and Combination of Errors Physics Notes

Absolute error: Absolute error in the measurement of a physical quantity is the magnitude of the difference between the true value and the measured value of the quantity.

Let a physical quantity can be measured n times.
Let the measured value be a₁, a₂, a₃, ………….a_n the arithmetic mean of these values is:
a_m = \(\frac{a_{1}+a_{2}+a_{3} \ldots a_{n}}{n}\)
Usually am is taken as the true value of the quantity, if the same is unknown otherwise by definition, absolute errors in the measured value of the quantity are

Absolute errors may be positive in certain cases and negative in other cases.

Mean absolute error: It is the arithmetic mean of the magnitude of absolute errors in all the measurements of the quantity it is represented by Aa. Thus,

Hence the final result of the measurement may be written as a = am ± Δ\(\bar{a}\).
This implies that any measurement of the quantity is likely to lie between (am+Δ\(\bar{a}\)) and (am – Δ\(\bar{a}\)).

Relative error: The relative error or fractional error of measurement is defined as the ratio of mean absolute error to the mean or value of the quantity measured. Thus,
Relative error = \(\frac{\text { Mean absolute error }}{\text { Mean value }}=\frac{\Delta \bar{a}}{a_{m}}\)

Percentage error: When the relative/fractional error is expressed in term of percentage, we call it percentage error. Thus,
Percentage error = \(\frac{\Delta \bar{a}}{a_{m}}\) × 100%

Combination of Errors:
Now, we will study how errors get combined while performing mathematical operations such as addition, subtraction, multiplication and division.
1. Error in Addition of Quantities
Let x = a + b
Let Δa = absolute error in measurements of a
Δb = absolute error in measurement of b
Δx = absolute error in addition of quantities
So, x ± Δx = (a ± Δa) + (b ± Δb)
or x ± Δx = (a + b) ± Δa ± Δb
or x ± Δx = x ± Δa ± Δb
or ±Δx = ± Δa ± Δb
Here, Δx can have four possible values:
(+Δa + Δb), (+Δa – Δb), (-Δa + Δb), (-Δa – Δb)
Maximum possible error in x Δx = ±(Δa + Δb)

2. Error in Difference of Quantities
Let x = a – b
Let Δa = absolute error in a
Δb = absolute error in b
Δx = absolute error in x
x ± Δx = (a ± Δ a) – (d ± Δ b)
or, x ± Δx = (a – d) ± Δa + Δd
or, x±Δx = x±Δa + Δb
or ±Δx = ±Δa + Δb
Here, Δx can have four possible values:
(+Δa – Δb), (-Δa + Δb), (-Δa – Δb), (+Δa + Δb)

Maximum absolute error indifference of quantities
Δx = ± (Δa + Δb)

3. Error in Relation of Multiplication of Physical Quantities
Let x = a x b
Let Δa = absolute error in measurements of a
Δb = absolute error in measurements of b
Δx = absolute error in the product of a and b
x ± Δx = (a ± Δa) x (b ± Δb)

The values of both \(\frac{\Delta a}{a}\) and \(\frac{\Delta b}{b}\) is very small. So, their product will be very very small. Hence, it can be neglected.

4. Error in Relation of Physical Quantities
Let x = \(\frac{a}{b}\)
Let Δa = absolute error in a
Δb = absolute error in b

Δx = absolute error in \(\frac{a}{b}\)

The values of both \(\frac{\Delta a}{a}\) and \(\frac{\Delta b}{b}\) is very small. So, the value of their product will be very small and it can be neglected.

5. Error Due to the Power of Physical Quantities

Physics Notes

Accuracy and Errors in Measurement Physics Notes

Accuracy and Errors in Measurement:
The measuring process is essentially a process of comparison. To measure any physical quantity, we compare it with a standard (unit) of that quantity. No measurement is perfect as the errors involved in the process cannot be removed completely. Hence, in spite of our best effort, the measured value of a quantity is always somewhat different from its actual value, or true value.

Measurement error: The measurement error is defined as the difference between the true or the actual value and the measured value

i. e., the error is quantity = (True value – measurement value) of the quantity.

Types of errors in measurement:
The errors may arise from the different sources and are usually classified into the following types. These types are:
1. Systematic errors
2. Random errors
3. Gross errors

Their types are explained below in detail.
(A) Systematic errors: Systematic errors are those which occur according to a definite pattern. These errors affect the measurement alike i. e., in the same way. The causes of systematic errors are known. Therefore, such errors can be minimized.

Some of the sources of systematic errors are:
1. Instrumental errors: These errors mainly arise due to the three reasons:
(a) Inherent shortcomings of the instrument: Such types of errors are inbuilt in the instruments because of their mechanical structure. They are due to the manufacturing, calibration or operation of the device. These errors may cause the error to read too low or too high.

For example: If the instrument uses the weak spring then it gives the high value of measuring quantity. The error occurs in the instrument because of the friction or hysteresis loss.

(b) Misuse of the instrument: The error occurs in the instrument because of the fault of the operator. A good instrument used in an unintelligent way may give an enormous result.

For example: The misuse of the instrument may cause the failure to adjust the zero of instruments, poor initial adjustment, using leads of high resistance. These improper practices may not cause permanent damage to the instrument, but all the same, they cause errors.

(c) Loading effect: It is the most common type of error which is caused by the instrument in the measurement work. For example, when the voltmeter is connected to the high resistance circuit it gives a misleading reading, and when it is connected to the low resistance circuit, it gives the dependable reading. This means the voltmeter has a loading effect on the circuit.

2. Natural errors: These errors are due to the external conditions of the measuring devices. Such types of errors mainly occur due to the effect of temperature, humidity, dust, vibration or because of the magnetic or electrostatic field. The corrective measures employed to eliminate or to reduce these undesirable effects are:
The arrangement should be made to keep the conditions as constant as pcoiible.
Using the equipment which is free from these effects.
By using the techniques which eliminate the effect of these disturbances.
By applying the computed corrections.

3. Observation error: Such type of errors are due to the wrong observation of the reading. There are many sources of observational error. For example, the pointer of a voltmeter reset slightly above the surface of the scale. Thus, an error occurs unless the line of vision of the observer is exactly above the pointer. To minimise the parallax error highly accurate meters are provided with mirrored scales.

(B) Random errors: These errors are due to the unknown causes and are sometimes termed as chance errors. In an experiment, even the same person repeating an observation may get different reading every time. For example, measuring diameter of a wire with a screw gauge, one may get different readings in different observations. It may happen due to many reasons. For example due to non-uniform area of cross-section of the wire at different places, the screw might have been tightened unevenly in different observation, etc. In such a case, it may not be possible to indicate which observation is most accurate. However, if we repeat the observation a number of times, the arithmetical mean of all the reading is found to be most accurate or very close to the most accurate reading for that observation. That is why, for an experiment it is recommended to repeat an observation a number of times and then to take their arithmetical mean.

If a₁, a₂, a₃………., a_n are the n different readings in an experiment, their arithmetic mean is given by.

(C) Gross errors: This error occurs because of human mistakes. For example consider the person using the instrument takes the wrong reading, or they can record the incorrect data. Such type of error comes under the gross error. The gross error can only be avoided by taking the reading carefully.

Example: The experimenter reads the reading 31.5°C as while the actual reading is 21.5°C. This happens because of the oversights. The experimenter takes the wrong reading and because of which the error occurs in measurement.

Such type of errors are very common in measurement. The complete elimination of such type of errors is not possible. Some of the gross errors are easily detected by the experimenter but some of them are difficult to find out.

Two methods can be used to remove the gross error:

1. The reading should be taken out very carefully.
2. Two or more readings should be taken of the measurement quantity. The readings are taken by the different experimenter and at a different point to remove the errors.

Physics Notes