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Thought-Process to Discover Knowledge

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mathsVector AlgebraVector Dot Product

### Vector Dot Product : Component Form

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Given the two vectors
vec p = p_x i+p_yj+p_zk

vec q = q_x i+q_yj+q_zk
what will be the vector dot product?

• |p||q|cos theta
• |p||q|cos theta
• |p||q|

answer is |p||q|cos theta.

How will one compute the angle theta from the given component forms of vectors
vec p = p_x i+p_yj+p_zk
vec q = q_x i+q_yj+q_zk

Consider this as triangle in coordinate plane.
vec p = p_x i+p_yj+p_zk
vec q = q_x i+q_yj+q_zk. The triangle is made of 3 sides having scalar quantities p=|vec p|, q=|vec q| and r = |vec q-vec p|. The cosine rule of triangle is applicable.
r^2=p^2+q^2-2pq cos theta

With algebraic manipulations on this, we can derive that
cos theta = (p_xq_x+p_yq_y+p_zq_z)/(|p||q|)
Substituting the above in the vector dot product we get.
vec p cdot vec q
quad quad = |p||q|cos theta
quad quad = |p||q|(p_xq_x+p_yq_y+p_zq_z)/(|p||q|)
quad quad = p_xq_x+p_yq_y+p_zq_z

That derives the component form of vector dot product as
vec p cdot vec q
quad quad = p_xq_x+p_yq_y+p_zq_z

This proof requires one to recall the cosine rule of triangles. A simpler proof, that a student can easily derive, is given in the coming pages.

Bilinear Property : For any vector vec p, vec q, vec r in bbb V and lambda in RR
(lambda vec p + vec q) cdot vec r = lambda (vec p cdot vec r) + (vec q cdot vec r)
This is explained and proven in properties of the dot product. For now, consider this to be true.

A vector vec p = p_x i + p_y j + p_z k is sum of scalar multiple of vectors. i, j, k are unit vectors, and the scalar multiples are p_x, p_y, p_z.

The same applies to vec q = q_x i + q_y j + q_z k Sum of multiple of vectors.

Proof for component form of vector dot product using bilinear property of dot product.

vec p cdot vec q
quad quad = (p_x i + p_y j + p_z k) cdot
quad quad quad quad (q_x i + q_y j + q_z k)
Apply bilinear property of dot product
quad quad = p_x i cdot (q_x i + q_y j + q_z k)
quad quad quad quad + p_y j cdot (q_x i + q_y j + q_z k)
quad quad quad quad + p_z k cdot (q_x i + q_y j + q_z k)
Apply i cdot i = i,j cdot j = j, k cdot k = k
i cdot j = 0, j cdot k = 0, k cdot i = 0
quad quad = p_xq_x+p_yq_y+p_zq_z

For given two vectors in component forms, the dot product is the sum of product of corresponding components of the vectors.

Vector Dot Product in Component Form: For given two vectors vec p = p_x i+p_yj+p_zk and
vec q = q_x i+q_yj+q_zk

vec p cdot vec q = p_xq_x+p_yq_y+p_zq_z

Solved Exercise Problem:

Given vec p = 2i+1.2j-k and vec q = i-j+k what is vec p cdot vec q?

• 2.2
• .2
• -.2
• -.2
• 4.2

The answer is '-.2'

slide-show version coming soon