Question

two vectors A=2i+3j-4k and B=4i+8j+xk are such that component of B along A is zero. then x is

Answer 1

component of $\vec{B}$ along $\vec{A}$ is $\frac{\vec{A}.\vec{B}}{|A|}$

here given,

A = 2i + 3j - 4k and B = 4i + 8j + xk

so, dot product of A and B = A.B = (2i + 3j - 4k).(4i + 8j + xk)

= 2 × 4 + 3 × 8 + (-4) × x

= 8 + 24 - 4x

= 32 - 4x

and magnitude of A = |A| = √{2² + 3² + 4²}

= √{4 + 9 + 16}

= √29

so, component of B along A = (32 - 4x)/√29.

according of question, component of B along A is zero.

so, (32 - 4x)/√29 = 0

or, 32 - 4x = 0

or, x = 32/4 = 8

hence, answer is x = 8