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Question #67228

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors such that their scalar triple product is $V$ . Let $A$ be a $3 \times 3$ matrix whose columns are the components of $\vec{a}, \vec{b}, \vec{c}$ respectively. Let the inverse of matrix $A$ be denoted by $A^{-1} = \begin{pmatrix} p_1 & p_2 & p_3 \\ q_1 & q_2 & q_3 \\ r_1 & r_2 & r_3 \end{pmatrix}$ . If $\Pi$ is a plane given by the equation $p_1 x + p_2 y + p_3 z + 1 = 0$ , then the position vector of the image of the origin with respect to the plane $\Pi$ is:

Options

A.
$-\frac{2V}{|\vec{b} \times \vec{c}|^2} (\vec{b} \times \vec{c})$
B.
$\frac{2V}{|\vec{b} \times \vec{c}|^2} (\vec{b} \times \vec{c})$
C.
$-\frac{V}{|\vec{b} \times \vec{c}|^2} (\vec{b} \times \vec{c})$
D.
$-\frac{2V}{|\vec{a} \times \vec{b}|^2} (\vec{a} \times \vec{b})$

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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

Inversion and Determinant Product|Scalar Triple Product Volume Logic|Image of a point in a plane

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 matrix inversion, reciprocal vector systems, and 3D coordinate geometry.

The primary trap lies in the sign convention of the plane equation derived from the inverse matrix.

Students often incorrectly assume the image formula without accounting for the $+1$ constant, leading to a missing negative sign.

Additionally, confusing the foot of the perpendicular with the image leads to a missing factor of 2.

Finally, it requires recognizing that the rows of the inverse of a column-vector matrix form the reciprocal vector system.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Inversion and Determinant Product
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free