Let a⃗ = î + ĵ - k̂ and c⃗ = 2î - 3ĵ + 2k̂. Then the number of vectors b⃗ such that b⃗ × c⃗ = a⃗ and |b⃗ | ∈ 1, 2, ........, 10 is :

Solution

Maths sir · 3 steps

Check a⃗ · c⃗ first: anything crossed with c⃗ is perpendicular to c⃗.

1. Cross product property. For any vector b⃗, the cross product b⃗ × c⃗ is perpendicular to c⃗. So (b⃗ × c⃗) · c⃗ = 0 always.
2. Condition on a. If b⃗ × c⃗ = a⃗, then a⃗ must also be perpendicular to c⃗. That means a⃗ · c⃗ = 0 is required before any b⃗ can exist.

   a⃗ · c⃗ = (1)(2) + (1)(-3) + (-1)(2) = 2 - 3 - 2 = -3

3. So. Since a⃗ · c⃗ = -3 ≠ 0, no vector b⃗ of any length satisfies b⃗ × c⃗ = a⃗. The number of such vectors is 0, option (a).

The idea to keep. The equation b⃗ × c⃗ = a⃗ has a solution only when a⃗ · c⃗ = 0. Check this one dot product first, before doing any component work.

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