Let a⃗=î+2 ĵ+3 k̂, b⃗=î-ĵ+2 k̂ and c⃗=5 î-3 ĵ+3 k̂ be three vectors. If r⃗ is a vector such that, r⃗ × b⃗=c⃗ × b⃗ and r⃗ · a⃗=0, then 25|r⃗|² is equal to :

Solution

Maths sir · 4 steps

Same cross product with b⃗ means r⃗=c⃗+λb⃗; the dot condition fixes λ.

1. Parallel to b. From r⃗×b⃗=c⃗×b⃗ we get (r⃗-c⃗)×b⃗=0⃗. A zero cross product means the two vectors are parallel, so r⃗=c⃗+λb⃗ for some scalar λ.
2. Use the dot condition. Dot both sides with a⃗ and set r⃗·a⃗=0. Here c⃗·a⃗=5-6+9=8 and b⃗·a⃗=1-2+6=5.

   0=c⃗·a⃗+λ (b⃗·a⃗)=8+5λ ⇒ λ=-8/5

3. Find r. r⃗=(5î-3ĵ+3k̂)-8/5(î-ĵ+2k̂)=1/5(17î-7ĵ-k̂).
4. So. |r⃗|²=(289+49+1)/25=339/25, so 25|r⃗|²= 339, which is option (c).

The idea to keep. Whenever you see r⃗×b⃗=c⃗×b⃗, write r⃗=c⃗+λb⃗ at once. Then the extra dot-product condition fixes λ.

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