Question

If $\bar{p} = 2\hat{\text{i}} + \hat{\text{k}}, \bar{q} = \hat{\text{i}} + \hat{\text{j}} + \hat{\text{k}}, \bar{r} = 4\hat{\text{i}} - 3\hat{\text{j}} + 7\hat{\text{k}}$ and a vector $\bar{\text{m}}$ is such that $\bar{\text{m}} \times \bar{q} = \bar{r} \times \bar{q}, \bar{\text{m}} \cdot \bar{p} = 0$, then $\bar{\text{m}} = .......$

• A. $\hat{\text{i}} - 8\hat{\text{j}} - 2\hat{\text{k}}$
• B. $-10\hat{\text{i}} + 3\hat{\text{j}} + 7\hat{\text{k}}$
• C. $-\hat{\text{i}} - 8\hat{\text{j}} + 2\hat{\text{k}}$
• D. $2\hat{\text{i}} + 4\hat{\text{j}} + \hat{\text{k}}$

Hint:

$\vec{a} \times \vec{b} = \vec{c} \times \vec{b} \implies \vec{a} - \vec{c}$ is parallel to $\vec{b}$.

Answer 1

The Correct Option is C

Step 1: Simplify Cross Product
$(\bar{m} - \bar{r}) \times \bar{q} = 0 \implies \bar{m} - \bar{r} = \lambda \bar{q}$.
$\bar{m} = \bar{r} + \lambda \bar{q} = (4+\lambda)\hat{i} + (-3+\lambda)\hat{j} + (7+\lambda)\hat{k}$.
Step 2: Use Dot Product
$\bar{m} \cdot \bar{p} = 0 \implies [(4+\lambda)\hat{i} + (-3+\lambda)\hat{j} + (7+\lambda)\hat{k}] \cdot (2\hat{i} + \hat{k}) = 0$.
$2(4+\lambda) + 1(7+\lambda) = 0 \implies 8 + 2\lambda + 7 + \lambda = 0 \implies 3\lambda = -15 \implies \lambda = -5$.
Step 3: Find $\bar{m$}}
$\bar{m} = (4-5)\hat{i} + (-3-5)\hat{j} + (7-5)\hat{k} = -\hat{i} - 8\hat{j} + 2\hat{k}$.
Final Answer: (C)