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 \(3a+2b = 7\), the ratio of a to b is \(x/2\). The value of x is

• A. \(4\)
• B. \(1\)
• C. \(8\)
• D. \(3\)

Hint:

Perpendicular vectors have zero dot product; solve the two equations for a and b.

Answer 1

The Correct Option is B

Step 1: Eliminate a
Multiply $3a+2b=7$ by 2 and $2a-3b=-4$ by 3: $6a+4b = 14$ and $6a - 9b = -12$.

Step 2: Subtract
$13b = 26$, so $b = 2$. Then $3a = 7-4 = 3$, so $a = 1$.

Step 3: Ratio
$a/b = 1/2$, which equals $x/2$ for $x=1$. Option (B).

Final Answer:
x = 1. \[ \boxed{\text{(B)}\ 1} \]