Question

The straight line $\vec{r} = (\hat{i} + \hat{j} + 2\hat{k}) + t(2\hat{i} + 5\hat{j} + 3\hat{k})$ is parallel to the plane $\vec{r} \cdot (2\hat{i} + \hat{j} - 3\hat{k}) = 5$. Then the distance between the straight line and the plane is:

• A. $\frac{9}{\sqrt{14}}$
• B. $\frac{8}{\sqrt{14}}$
• C. $\frac{7}{\sqrt{14}}$
• D. $\frac{6}{\sqrt{14}}$
• E. $\frac{5}{\sqrt{14}}$

Hint:

Verify parallelism first by checking $\vec{b} \cdot \vec{n} = 0$. Here, $(2, 5, 3) \cdot (2, 1, -3) = 4 + 5 - 9 = 0$. This confirms the line is parallel and the distance is constant.

Answer 1

The Correct Option is B