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Question #75695

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(\vec{a})$ and $B(\vec{b})$ be two distinct points in space, and $\Pi$ be a plane passing through the origin with a unit normal vector $\hat{n}$ . A point $C$ divides the line segment $AB$ internally in the ratio $k:1$ . The ray $AC$ is orthogonally projected onto the plane $\Pi$ to form a ray $A'R$ , where $A'$ is the orthogonal projection of $A$ on $\Pi$ . A point $Q$ lies on the ray $A'R$ such that the geometric length $A'Q$ is equal to the length of the segment $AC$ . If $\vec{v} = \vec{b} - \vec{a}$ and $\vec{v} \times \hat{n} \neq \vec{0}$ , which of the following represents the position vector of $Q$ ?

Options

A.
$\vec{a} - (\vec{a} \cdot \hat{n})\hat{n} + \frac{k}{k+1} \frac{|\vec{v}|}{|\vec{v} \times \hat{n}|} (\vec{v} - (\vec{v} \cdot \hat{n})\hat{n})$
B.
$\vec{a} - (\vec{a} \cdot \hat{n})\hat{n} + \frac{k}{k+1} (\vec{v} - (\vec{v} \cdot \hat{n})\hat{n})$
C.
$\vec{a} - (\vec{a} \cdot \hat{n})\hat{n} + \frac{k}{k+1} \frac{1}{|\vec{v} \times \hat{n}|} (\vec{v} - (\vec{v} \cdot \hat{n})\hat{n})$
D.
$\vec{a} - (\vec{a} \cdot \hat{n})\hat{n} + \frac{k}{k+1} \frac{|\vec{v} \times \hat{n}|}{|\vec{v}|} (\vec{v} - (\vec{v} \cdot \hat{n})\hat{n})$

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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

Scalar Multiplication Geometric Scaling|Orthogonal projection of a line|Ratio of Division (k:1)

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 candidate's ability to distinguish between the geometric scaling of a vector during orthogonal projection and the parametric partition of a segment using the section formula.

The trap lies in Option B, which represents the direct orthogonal projection of point C onto the plane.

Students commonly assume that applying the section formula ratio $k:1$ to the projected endpoints $A'$ and $B'$ will yield a point at the required absolute distance.

However, projection scales the magnitude of the direction vector by a factor of $\sin \theta$ (or $|\vec{v} \times \hat{n}|/|\vec{v}|$ ).

To find a point on the projected ray with a specific absolute length, one must explicitly normalize the projected direction vector and multiply it by the original segment's length, rather than relying on the unnormalized affine transformation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Scalar Multiplication Geometric Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free