Let a⃗=î+4 ĵ+2 k̂, b⃗=3 î-2 ĵ+7 k̂ and c⃗=2 î-ĵ+4 k̂. If a vector d⃗ satisfies d⃗ × b⃗=c⃗ × b⃗ and d⃗ · a⃗=24, then |d⃗|² is equal to :

Solution

Maths sir · 4 steps

A zero cross product means parallel: d⃗-c⃗ is a multiple of b⃗.

1. Parallel to b. From d⃗×b⃗=c⃗×b⃗ we get (d⃗-c⃗)×b⃗=0⃗. A zero cross product means d⃗-c⃗ is parallel to b⃗. So d⃗=c⃗+λb⃗ for some number λ.
2. Find lambda. Take the dot product with a⃗. Here c⃗·a⃗=2-4+8=6 and b⃗·a⃗=3-8+14=9.

   d⃗·a⃗=c⃗·a⃗+λ b⃗·a⃗ ⇒ 6+9λ=24 ⇒ λ=2

3. Build d. d⃗=c⃗+2b⃗=(2+6)î+(-1-4)ĵ+(4+14)k̂=8î-5ĵ+18k̂.
4. So. |d⃗|²=8²+(-5)²+18²=64+25+324= 413, which is option (b).

The idea to keep. Whenever you see x⃗×b⃗=y⃗×b⃗, write x⃗=y⃗+λb⃗. Then one dot-product condition is enough to fix λ.

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