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Question #76005

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\theta$ , where $\theta \in \left(\frac{\pi}{2}, \pi\right)$ . Let $P_1$ be the plane passing through the origin, containing the vector $\vec{a}$ , and perpendicular to the plane containing $\vec{a}$ and $\vec{b}$ . Let $P_2$ be the plane passing through the origin, containing the vector $\vec{b}$ , and perpendicular to the plane containing $\vec{a}$ and $\vec{b}$ . If $\alpha$ is the acute angle between the planes $P_1$ and $P_2$ , then $\cos \alpha$ is equal to:

Options

A.
$-\cos \theta$
B.
$\cos \theta$
C.
$-\sin \theta$
D.
$\sin \theta$

Check Your Answer

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

Vector Triple Product Expansion|Angle between two planes|Symmetric vs Non-symmetric form|ASTC Quadrant Sign Convention

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the intersection of Vector Triple Product (VTP) expansions and 3D geometry, specifically targeting the definition of the angle between two planes.

Students must first correctly apply the VTP to find the normal vectors of planes defined in non-symmetric form.

The primary trap lies in the ASTC quadrant sign convention combined with the definition of the acute angle between planes.

The dot product of the normals yields $\cos \phi = \cos \theta$ .

Because $\theta \in (\pi/2, \pi)$ , $\cos \theta$ is negative, meaning the angle between the normals is obtuse.

Students who blindly equate the angle between normals to the angle between planes will incorrectly choose $\cos \theta$ .

The true acute angle $\alpha$ requires taking the absolute value, yielding $\cos \alpha = -\cos \theta$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Vector Triple Product Expansion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free