a ⃗=3i ̂+2j ̂+2k ̂ and b ⃗=i ̂+2j ̂-2k ̂ be two vectors. If a vector perpendicular to both vectors
Let a ⃗=3i ̂+2j ̂+2k ̂ and b ⃗=i ̂+2j ̂-2k ̂ be two vectors. If a vector perpendicular to both the vectors a ⃗+b ⃗ and a ⃗-b ⃗ has the magnitude 12 then one such vector is
(a) 4(2i ̂-2j ̂-k ̂ ) (b) 4(-2i ̂-2j ̂+k ̂ )
(c) 4(2i ̂+2j ̂+k ̂ ) (d) 4(2i ̂+2j ̂-k ̂ )
Ans: a
Sol.
Required vector r ⃗=λ(a ⃗×b ⃗ )
[∵r ⃗ is perpendicular to both a ⃗+b ⃗ and a ⃗-b ⃗]
⇒r ⃗=λ|■(i ̂&j ̂&k ̂@3&2&2@1&2&-2)|⇒r ⃗=λ(-8i ̂+8j ̂+4k ̂ )
⇒r ⃗=-4λ(2i ̂-2j ̂-k ̂ )
Also, |r ⃗ |=12 [Given]
∴|4λ| √(4+4+1)=12⇒3|4λ|=12
⇒λ=±1
∴ Required vector r ⃗=±4(2i ̂-2j ̂-k ̂ )
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