Question

Two vectors \(a\hat{i} + b\hat{j} + \hat{k}\) and \(2\hat{i} - 3\hat{j} + 4\hat{k}\) are perpendicular. When \(3a + 2b = 7\) , the ratio \(a/b\) is \(x/2\). The value of \(x\) is.

• A. \(\text{zero}\)
• B. \(2\)
• C. \(1\)
• D. \(4\)

Hint:

Dot product of perpendicular vectors is always zero.

Answer 1

The Correct Option is C

Step 1: Dot Product
Since vectors are perpendicular, \((a)(2) + (b)(-3) + (1)(4) = 0\).
\(2a - 3b = -4\).

Step 2: Solve System
1) \(2a - 3b = -4\)
2) \(3a + 2b = 7\)
Multiplying (1) by 2 and (2) by 3:
\(4a - 6b = -8\)
\(9a + 6b = 21\)
Adding: \(13a = 13 \implies a = 1\).
Substituting \(a=1\): \(2(1) - 3b = -4 \implies 3b = 6 \implies b = 2\).

Step 3: Ratio
\(a/b = 1/2\). Given ratio is \(x/2\), so \(x = 1\).
Final Answer: (C)