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: Understanding the Question:
Two vectors are perpendicular, meaning their dot product is zero. We also have a linear equation relating the coefficients. We need to solve for the coefficients and find a ratio.
Step 3: Detailed Explanation:
Let \(\vec{u} = a\hat{i} + b\hat{j} + \hat{k}\) and \(\vec{v} = 2\hat{i} - 3\hat{j} + 4\hat{k}\).
Since \(\vec{u} \perp \vec{v}\), \(\vec{u} \cdot \vec{v} = 0\):
\[ (a)(2) + (b)(-3) + (1)(4) = 0 \]
\[ 2a - 3b + 4 = 0 \implies 2a - 3b = -4 \quad \dots \text{(i)} \]
We are also given:
\[ 3a + 2b = 7 \quad \dots \text{(ii)} \]
Solve the system of equations. Multiply (i) by 2 and (ii) by 3:
\[ 4a - 6b = -8 \]
\[ 9a + 6b = 21 \]
Adding the two equations:
\[ 13a = 13 \implies a = 1 \]
Substitute \(a = 1\) into equation (ii):
\[ 3(1) + 2b = 7 \implies 2b = 4 \implies b = 2 \]
The ratio of \(a\) to \(b\) is:
\[ \frac{a}{b} = \frac{1}{2} \]
According to the problem, this ratio is \(\frac{x}{2}\). Therefore:
\[ \frac{x}{2} = \frac{1}{2} \implies x = 1 \]
Step 4: Final Answer:
The value of \(x\) is \(1\).