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: Understanding the Concept:
$\bar{m} \times \bar{q} = \bar{r} \times \bar{q} \implies (\bar{m} - \bar{r}) \times \bar{q} = 0$.
This implies $\bar{m} - \bar{r}$ is parallel to $\bar{q}$.
Step 2: Key Formula or Approach:
$\bar{m} = \bar{r} + \alpha \bar{q}$.
Use $\bar{m} \cdot \bar{p} = 0$ to solve for $\alpha$.
Step 3: Detailed Explanation:
$\bar{m} = (4+\alpha, -3+\alpha, 7+\alpha)$.
$\bar{m} \cdot (2, 0, 1) = 0 \implies 2(4+\alpha) + (7+\alpha) = 0$.
$8 + 2\alpha + 7 + \alpha = 0 \implies 3\alpha = -15 \implies \alpha = -5$.
$\bar{m} = (4-5, -3-5, 7-5) = (-1, -8, 2)$.
Step 4: Final Answer:
The vector is $-\hat{\text{i}} - 8\hat{\text{j}} + 2\hat{\text{k}}$.