Multiplication of vector by a scalar

Let is multiplied by a scalar m. If m is a positive quantity, only magnitude of the vector will change by a factor of ‘m’ and its direction will remain same. If m is a negative quantity the direction of the vector will be reversed, but magnitude of the vector remain same.

Multiplication of Vectors

Vector multiplications are of two types. One, in which we obtain a scalar quantity and the other in which we obtain a vector quantity on multiplication. The first one is called Dot Product and the other is called Cross Product.

(I) Scalar product: cosθ where θ is the angle between the two vectors, when placed tail to tail.

For θ = 90^0, cos θ = 0 then = 0

Now for orthogonal unit vectors, = 0

Again for θ = 0⁰, cosθ = 1 then = AB

Dot product is commutative. i.e.

Angle between two vectors:

As we know

⇒ cos θ

Illustration 13: Two constant forces newton and newton act together on a particle during its displacement from the position m to m. Calculate the work done. If work done by a force, for a displacement is given by.

Solution: Total work done W

= [+]

=

Illustration 14. Show that if, then the sum and difference of vectors and are at right angles.

Solution:

= 0 (Q )

Illustration 15. If and, find the angle between.

Solution: We have

cos θ =

Where θ is the angle between and

Now . = axbx + ayby + azbz

= 2 × 4 + 3 × 3 + 4 × 2 = 25

Also, a =

b =

Thus cos θ = or θ = cos-1 ()

(II) Cross product:

The cross product of two vectors inclined to each other by an angle θ is given by

The vector product is not commutative i.e. and

In terms of orthogonal vectors

,

Then, =

Cross Product of two parallel or anti parallel vectors is zero.

Note: Division of a vector by a vector is not defined.

Illustration 16. Find the cross product of two vectors and which are

(i) parallel and equal (ii) antiparallel and negative

Solution: If is parallel (or equal) to, θ = 00.

Hence | × | = AB sin0⁰ = 0

If is antiparallel (or negative) to, θ = 1800. Hence

| × | = AB sin180⁰ = 0

If is perpendicular to, θ = 90⁰

Then = | × | = AB sin90⁰ = AB and C(=×) is directed outwad obeying right hand thrum rule. × must be perpendicular to the plane containing and.

Illustration 17. If the magnitudes of the dot product and cross product of two vectors are equal, find the angle between the two vectors.

Solution: |

A.B sinθ = A.B cosθ ⇒ tanθ = 1

⇒ θ = 45⁰.