Let a⃗ = 2î - 3ĵ + 4k̂ and b⃗ = 7î + ĵ - 6k̂.
If r⃗ × a⃗ = r⃗ × b⃗, r⃗ . (î + 2ĵ + k̂) = -3, then r⃗ . (2î - 3ĵ + k̂) is equal to :

Solution

Maths sir · 4 steps

Move both cross products to one side, and r⃗ is forced to be parallel to a⃗ - b⃗.

1. Parallel vectors. Moving everything to one side gives r⃗ × (a⃗ - b⃗) = 0⃗. A cross product of two nonzero vectors is zero only when they are parallel. So r⃗ = λ(a⃗ - b⃗) for some scalar λ.
2. Find a minus b. a⃗ - b⃗ = (2-7)î + (-3-1)ĵ + (4+6)k̂ = -5î - 4ĵ + 10k̂.
3. Fix lambda. Use the given dot product: λ(-5 - 8 + 10) = -3, so -3λ = -3 and λ = 1. This gives r⃗ = -5î - 4ĵ + 10k̂.

   r⃗ · (2î - 3ĵ + k̂) = (-5)(2) + (-4)(-3) + (10)(1) = -10 + 12 + 10

4. So. The required dot product is 12, which is option (d).

The idea to keep. When r⃗ × a⃗ = r⃗ × b⃗, write r⃗ = λ(a⃗ - b⃗) and let the extra dot-product condition fix λ.

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