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Question #41927

Difficulty: 8/10 | 🟢 Free

The Question

Consider three points $A(2, -1, 4)$ , $B(-2, 3, -3)$ , and $C(-1, 1, -5)$ in 3D space. Let $p$ be the scalar projection of vector $\vec{BA}$ on vector $\vec{BC}$ , and $q$ be the scalar projection of vector $\vec{BC}$ on vector $\vec{BA}$ . If $\{x\}$ denotes the fractional part function, find the value of $18 \{ p \} + 9 \{ q \}$ .

Options

A.
13
B.
14
C.
17
D.
19

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Dot Product Projection Sign|Angle between two lines|Modulus Sign Dependency|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the precise definition of scalar projection and the fractional part function.

The 'Lethal Collision' occurs because the angle between the vectors is obtuse, yielding a negative dot product and negative scalar projections.

Students frequently fall into the trap of either taking the absolute value of the projection (ignoring the obtuse angle, leading to 14) or incorrectly computing the fractional part of a negative number as $\{-x\} = x$ , failing to use the strict definition $\{x\} = x - \lfloor x \rfloor$ .

Quick Facts About This Question

Question Information

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