3D Geometry
Distance from a point to a line
nta_pyq_2023_jan
Grade 12
Question:
The distance of the point P(4, 6, -2) from the line passing through the point $(-3, 2, 3)$ and parallel to a line with direction ratios $3, 3, -1$ is equal to:
3
\sqrt{6}
2\sqrt{3}
\sqrt{14}
Step-by-Step Solution
Key Concept: Use $d = \frac{|\overrightarrow{AP} \times \vec{b}|}{|\vec{b}|}$ where $A$ is the point on line and $\vec{b}$ is direction vector.
$\overrightarrow{AP} = (7,4,-5)$, $\vec{b}=(3,3,-1)$. $\overrightarrow{AP}\times\vec{b} = \hat{i}(-4+15)-\hat{j}(-7+15)+\hat{k}(21-12) = (11,-8,9)$. Wait — answer is $\sqrt{14}$. Correct: cross product gives magnitude $\sqrt{14 \cdot 19}/\sqrt{19} = \sqrt{14}$. Answer: (4)
Correct Answer: $\sqrt{14}$