Question

Let \( \vec{a} = 3\hat{i} + \hat{j} - 2\hat{k} \), \( \vec{b} = 4\hat{i} + \hat{j} + 7\hat{k} \), and \( \vec{c} = \hat{i} - 3\hat{j} + 4\hat{k} \) be three vectors.
If a vector \( \vec{p} \) satisfies \( \vec{p} \times \vec{b} = \vec{c} \times \vec{b} \) and \( \vec{p} \cdot \vec{a} = 0 \), then \( \vec{p} \cdot (\hat{i} - \hat{j} - \hat{k}) \) is equal to

• A. 24
• B. 36
• C. 28
• D. 32

Answer 1

The Correct Option is D

Given:

\[ \vec{p} \times \vec{b} - \vec{c} \times \vec{b} = 0 \quad \implies \quad (\vec{p} - \vec{c}) \times \vec{b} = 0 \]

This implies:

\[ \vec{p} - \vec{c} = \lambda \vec{b} \quad \implies \quad \vec{p} = \vec{c} + \lambda \vec{b} \]

Given that \( \vec{p} \cdot \vec{a} = 0 \), we have:

\[ (\vec{c} + \lambda \vec{b}) \cdot \vec{a} = 0 \]

Substituting values:

\[ \vec{c} \cdot \vec{a} + \lambda (\vec{b} \cdot \vec{a}) = 0 \] \[ (3 - 3 - 8) + \lambda (12 + 1 - 14) = 0 \quad \implies \quad \lambda = -8 \]

Thus:

\[ \vec{p} = \vec{c} - 8\vec{b} = -31\hat{i} - 11\hat{j} - 52\hat{k} \]

Now, compute:

\[ \vec{p} \cdot (\hat{i} - \hat{j} - \hat{k}) \] \[ = (-31)(1) + (-11)(-1) + (-52)(-1) \] \[ = -31 + 11 + 52 = 32 \]