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Question #40845

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$ . A vector $\vec{c}$ is collinear with $\vec{d} = 3\hat{i} + 2\hat{j} + \hat{k}$ . The volume of the parallelepiped formed by the vectors $\vec{a}, \vec{b},$ and $\vec{c}$ is $192$ cubic units. If the terminal point of $\vec{c}$ (when its initial point is at the origin) lies in the octant where all three coordinates are strictly negative, find the value of $[\vec{a} + \vec{b} \quad \vec{b} + \vec{c} \quad \vec{c} + \vec{a}]$ .

Options

A.
-384
B.
384
C.
-192
D.
192

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Octant Sign Convention|Collinearity and Coplanarity|Scalar Multiplication Geometric Scaling

Why This Question Is Tricky

Common Mistakes

The trap tests the student's ability to integrate geometric constraints (octant sign convention) with algebraic properties of the scalar triple product.

Students often mechanically equate the scalar triple product to the positive volume ( $[\vec{a}, \vec{b}, \vec{c}] = 192$ ) and apply the identity $[\vec{a}+\vec{b}, \vec{b}+\vec{c}, \vec{c}+\vec{a}] = 2[\vec{a}, \vec{b}, \vec{c}]$ to get $384$ .

However, the negative scalar multiplication ( $\lambda = -12$ ) geometrically scales and reflects the vector through the origin into the diametrically opposite octant.

This shift alters the orientation of the vector triad, making the signed volume (scalar triple product) strictly negative.

Failing to account for this orientation reversal leads to a lethal sign error.

Quick Facts About This Question

Question Information

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