Let ⃗a=−^i−^k, ⃗b=−^i+^j and ⃗c=^i+2^j+3^k be three given vectors. If ⃗r is a vector such that ⃗r×⃗b=⃗c×⃗b and ⃗r.⃗a=0, then the value of ⃗r.⃗b is<p></p>

Given ⃗r×⃗b=⃗c×⃗b ⇒(⃗r−⃗c)×⃗b=⃗0
So, ⃗r=⃗c+λ⃗b
⃗r=(1−λ)^i+(2+λ)^j+3^k
⃗r.⃗a=0 ⇒(1−λ)(−1)+3(−1)=0
or, −1+λ−3=0
or, λ=4
⇒⃗r= −3^i+6^j+3^k
⃗r.⃗b=(−3)(−1)+6(1)=3+6=9

Eduprobe

Question:

Let ⃗a=−^i−^k, ⃗b=−^i+^j and ⃗c=^i+2^j+3^k be three given vectors. If ⃗r is a vector such that ⃗r×⃗b=⃗c×⃗b and ⃗r.⃗a=0, then the value of ⃗r.⃗b is

Solution: