Question

Show that the scalar product of two vectors is equal to the sum of the products of their corresponding x,y,z components

Answer 1

┏_________∞◆∞_________┓

✭✮ӇЄƦЄ ƖƧ ƳƠƲƦ ƛƝƧƜЄƦ✮✭

┗_________∞◆∞_________┛

Let the 2 vectors be A and B

Scalar product of A and B = Dot product

A·B = ABcosФ

If,

A = ai + bj + ck

B = pi + qj + rk         where, i, j and k represent unit vectors along x, y and z

A·B = (ai + bj + ck)(pi + qj + rk)

In scalar product, we are taking the cosine component of the vectors

We know that,

cos0 = 1

cos90 = 0

When we do i·i or j·j or k·k ,

i·i = |i|×|i|×cos0 = 1×1×1 = 1 = j·j = k·k

When we do i·j or j·k or k·i ,

i·j = |i|×|j|×cos90 = 1×1×0 = 0 = j·k = k·i

Hence,

scalar product of two vectors is equal to the sum of the products of their corresponding x,y,z components