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Question #75902

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{p} = \alpha\hat{i} + 3\beta\hat{j} - \gamma\hat{k}$ and $\vec{q} = -2\alpha\hat{i} + \beta\hat{j} + 2\gamma\hat{k}$ be the position vectors of points $P$ and $Q$ respectively, where $\alpha, \beta, \gamma \in \mathbb{R}^+$ . Let $R(\vec{r})$ be the point dividing the line segment $PQ$ externally in the ratio $2:1$ . If $\vec{v} = x\hat{i} + y\hat{j} + z\hat{k}$ is a vector such that the scalar projection of $\vec{v}$ onto $\vec{r}$ is strictly negative for all possible values of $\alpha, \beta, \gamma$ , which of the following conditions must be satisfied by $x, y, z$ ?

Options

A.
$x > 0,\; y > 0,\; z < 0$
B.
$x > 0,\; y < 0,\; z < 0$
C.
$x < 0,\; y < 0,\; z > 0$
D.
$x < 0,\; y > 0,\; z > 0$

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

Section Formula Internals|Dot Product Projection Sign|Octant 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 student's ability to correctly apply the external section formula and track the resulting Octant Sign Convention of the position vector.

A common trap is mistakenly using the internal division formula, which yields a vector in a different octant $(-, +, +)$ , leading directly to Option B.

Another trap is reversing the ratio order ( $1:2$ instead of $2:1$ ), which yields $(+, +, -)$ and leads to Option C.

Finally, students must correctly interpret the dot product constraint for a negative projection, realizing that the signs of $x, y, z$ must oppose the signs of $\vec{r}$ 's components to guarantee a negative sum for all positive parameters.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Section Formula Internals
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free