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Question #40785

Difficulty: 8/10 | 🟢 Free

The Question

Let $P_1$ be a plane passing through the point $A(2, 2, \sqrt{2})$ with its normal vector parallel to $\hat{i} + \hat{j} + \sqrt{2}\hat{k}$ . Let $P_2$ be another plane passing through the point $B(-1, 1, -\sqrt{2})$ and perpendicular to the vector $\sqrt{2}\hat{i} - \sqrt{2}\hat{j} + 2\hat{k}$ . If $\vec{r}_1$ and $\vec{r}_2$ are the vectors along the directed perpendiculars drawn from the origin to the planes $P_1$ and $P_2$ respectively, then the angle between $\vec{r}_1$ and $\vec{r}_2$ is

Options

A.
$\frac{\pi}{3}$
B.
$\frac{2\pi}{3}$
C.
$\frac{\pi}{6}$
D.
$\frac{5\pi}{6}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Normal Form Parameters|Angle between two lines|Octant Sign Convention

Why This Question Is Tricky

Common Mistakes

This question tests the 'Octant Sign Convention' in the 3D normal form of a plane.

Students often calculate the angle between planes using the raw normal vectors derived from the Cartesian equations (which yields $\cos \theta = 1/2 \implies \pi/3$ ).

However, the directed perpendicular from the origin requires the plane equation to be normalized such that the distance parameter $p$ is strictly positive.

Failing to enforce $p > 0$ for $P_2$ results in the supplementary acute angle instead of the correct obtuse angle.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Normal Form Parameters
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free