If vectors a₁⃗ = xî - ĵ + k̂ and a₂⃗ = î + yĵ + zk̂ are collinear, then a possible unit vector parallel to the vector xî + yĵ + zk̂ is :

Solution

Maths sir · 3 steps

Write a₁⃗ = μ a₂⃗ and see which conditions it puts on x, y and z.

1. Use collinearity. Collinear vectors are scalar multiples of each other, so a₁⃗ = μ a₂⃗ for some μ ≠ 0. Comparing the î, ĵ and k̂ components gives x = μ, -1 = μ y and 1 = μ z.

   x = μ, y = -1/μ, z = 1/μ

2. Two conditions. From these, y = -z and xz = 1. A unit vector only fixes the direction, so in each option check two things: y = -z, and x and z have the same sign, so that xz > 0.

   Option	Direction (x, y, z)	y = -z?	xz > 0?
   (a)	(1, -1, 1)	yes	yes, xz = 1
   (b)	(0, -1, 1)	yes	no, x = 0
   (c)	(1, -1, 0)	no	no, z = 0
   (d)	(1, 1, -1)	yes	no, xz = -1

3. So. With μ = 1 you get x = 1, y = -1, z = 1. The vector is î - ĵ + k̂, with magnitude √3. The unit vector is 1/√3(î - ĵ + k̂), option (a).

The idea to keep. Collinear means one vector is a scalar times the other. Put in the multiplier μ, compare components, and read off the relations between the unknowns.

Answer from the NTA final answer key. Solution by PrepWiser.