Let a⃗=2 î-7 ĵ+5 k̂, b⃗=î+k̂ and c⃗=î+2 ĵ-3 k̂ be three given vectors. If r⃗ is a vector such that r⃗ × a⃗=c⃗ × a⃗ and r⃗ · b⃗=0, then |r⃗| is equal to :

Solution

Maths sir · 4 steps

Bring it to one side: a zero cross product means r⃗-c⃗ is parallel to a⃗.

1. Use the cross product. From r⃗×a⃗=c⃗×a⃗ we get (r⃗-c⃗)×a⃗=0⃗. A zero cross product means the two vectors are parallel, so r⃗=c⃗+λa⃗ for some scalar λ.
2. Use the dot product. Dot with b⃗ and use r⃗·b⃗=0. Here c⃗·b⃗=1-3=-2 and a⃗·b⃗=2+5=7.

   c⃗·b⃗+λ a⃗·b⃗=0 ⇒ -2+7λ=0 ⇒ λ=2/7

3. Build r. r⃗=(î+2ĵ-3k̂)+2/7(2î-7ĵ+5k̂)=11/7î+0 ĵ-11/7k̂.
4. So. |r⃗|=√((11/7)²+(11/7)²)=11/7√2, which is option (d).

The idea to keep. Whenever you see r⃗×a⃗=c⃗×a⃗, write r⃗=c⃗+λa⃗ at once. The second condition then fixes λ.

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