Representation of 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