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Question #40870

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(2, 0, 0)$ , $B(0, 3, 0)$ , and $C(0, 0, 6)$ be three points in $\mathbb{R}^3$ . Let $\vec{n}_1 = \vec{AC} \times \vec{AB}$ and $\hat{n}_1$ be the unit vector along $\vec{n}_1$ . Let $P_2$ be the plane $2x - y + 3z = 4$ with normal vector $\vec{n}_2 = 2\hat{i} - \hat{j} + 3\hat{k}$ and $\hat{n}_2$ be the unit vector along $\vec{n}_2$ . Let $\Pi$ be the plane bisecting the acute angle between the plane $ABC$ and $P_2$ . If $\vec{n}_3$ is the normal vector to $\Pi$ expressed in the form $\vec{n}_3 = \alpha \hat{n}_1 + \beta \hat{n}_2$ where $\alpha > 0$ , find the scalar projection of the vector $\vec{v} = 5\hat{i} - \hat{j} + 4\hat{k}$ onto $\vec{n}_3$ .

Options

A.
$-\frac{20\sqrt{42}}{21}$
B.
$\frac{20\sqrt{42}}{21}$
C.
$-\frac{10\sqrt{42}}{21}$
D.
$\frac{10\sqrt{42}}{21}$

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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

Coordinate Axis Projections|Angle between two planes|Vector Product Anti-Commutativity

Why This Question Is Tricky

Common Mistakes

This question tests the student's attention to the anti-commutativity of the vector cross product ( $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$ ) when defining plane normals, and its cascading effect on finding the acute angle bisector.

Students often habitually compute the normal as $\vec{AB} \times \vec{AC}$ regardless of the specified order $\vec{AC} \times \vec{AB}$ .

This sign error flips the sign of the dot product $\hat{n}_1 \cdot \hat{n}_2$ from negative to positive.

Consequently, they incorrectly select $\hat{n}_1 + \hat{n}_2$ instead of $\hat{n}_1 - \hat{n}_2$ for the acute bisector normal, leading to a sign reversal in the final scalar projection.

The trap exploits the assumption that normal vector orientation is arbitrary when bisecting planes.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Coordinate Axis Projections
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free