Question

Vectors $a \hat{i}+b \hat{j}+\hat{k}$ and $2 \hat{i}-3 \hat{j}+4 \hat{k}$ are perpendicular to each other when $3 a+2 b=7$, the ratio of $a$ to $b$ is $\frac{x}{2}$ The value of $x$ is __

Answer 1

The correct answer is 1.
For two perpendicular vectors
(ai^+bj^​+k^)⋅(2i^−3j^​+4k^)=0
2a−3b+4=0
On solving, 2a−3b=−4
Also given
3a+2b=7
We get a=1,b=2
ba​=2x​⇒x=b2a​=22×1​
⇒x=1