Product of Vectors Physics Notes

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 \(\vec{B}\)

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