Question

The velocity of a particle is 3i + 2j + 3k. The component of this velocity perpendicular to the line i-j+k is:
A) 5/3(-2i-j+k)

B) 4/3(i-j+k)

C) 5/3(i +2j + k)

D) 4/3(i+2j+k)

Answer 1

answer : option (C) 5/3(i + 2j + k)

explanation : concept : if a and b are two vectors , then component of a perpendicular to b = a - (a.b)b/|b|²

here, a = (3i +2j + 3k) and b = (i - j + k)

magnitude of b, |b| = √{1² + (-1)² + 1²} = √3

and dot product of a and b, a.b = (3i + 2j + 3k).(i - j + k) = 3 - 2 + 3 = 4

so, component of a perpendicular to b = (3i + 2j + 3k) - 4/(√3)² (i - j + k)

= (3i + 2j + 3k) - 4/3 (i - j + k)

= (3 - 4/3)i + (2 + 4/3)j + (3 - 4/3)k

= 5/3 i + 10/3 j + 5/3 k

= 5/3 (i + 2j + k)

hence, component of velocity of particle perpendicular to the line (i - j + k) is 5/3(i + 2j + k)