Let a⃗ = î + 2ĵ - k̂, b⃗ = î - ĵ and c⃗ = î - ĵ - k̂ be three given vectors. If r⃗ is a vector such that r⃗ × a⃗ = c⃗ × a⃗ and r⃗ . b⃗ = 0, then r⃗ . a⃗ is equal to __________.

Solution

Maths sir · 3 steps

Move both cross products to one side: r⃗ - c⃗ is parallel to a⃗.

1. Parallel to a. From r⃗ × a⃗ = c⃗ × a⃗ we get (r⃗ - c⃗) × a⃗ = 0⃗. A zero cross product means r⃗ - c⃗ is parallel to a⃗, so r⃗ = c⃗ + λ a⃗ for some scalar λ.
2. Use the dot condition. Dot both sides with b⃗: c⃗ · b⃗ + λ a⃗ · b⃗ = 0. Here c⃗ · b⃗ = 1 + 1 + 0 = 2 and a⃗ · b⃗ = 1 - 2 + 0 = -1, so 2 - λ = 0 and λ = 2.

   r⃗ = c⃗ + 2a⃗ = 3î + 3ĵ - 3k̂

3. So. r⃗ · a⃗ = (3)(1) + (3)(2) + (-3)(-1) = 3 + 6 + 3 = 12. As a check, r⃗ · b⃗ = 3 - 3 = 0.

The common slip: -12. Writing a⃗ · b⃗ as +1 instead of -1 gives λ = -2, so r⃗ = -î - 5ĵ + k̂ and r⃗ · a⃗ = -12.

The idea to keep. Whenever r⃗ × a⃗ = c⃗ × a⃗, write r⃗ = c⃗ + λ a⃗ at once. The second condition given is there only to fix λ.

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