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Question #40851

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}, \vec{b}, \vec{c}$ be three unit vectors such that the angle between any two of them is $\frac{\pi}{3}$ . Let $\vec{r} = \vec{a} - \vec{b} + 2\vec{c}$ . If $\vec{P}_1 = \vec{a} \times (\vec{r} \times \vec{a})$ , $\vec{P}_2 = \vec{b} \times (\vec{r} \times \vec{b})$ , and $\vec{P}_3 = \vec{c} \times (\vec{r} \times \vec{c})$ , then the volume of the parallelepiped determined by the vectors $\vec{P}_1, \vec{P}_2, \vec{P}_3$ is:

Options

A.
$\sqrt{2}$
B.
$\frac{1}{\sqrt{2}}$
C.
$2\sqrt{2}$
D.
$\frac{3}{\sqrt{2}}$

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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

Projection vs. Coordinate|Coplanarity Condition|Vector Triple Product Expansion

Why This Question Is Tricky

Common Mistakes

This problem tests the student's ability to handle vector projections within a non-orthogonal basis.

The 'Lethal Collision' occurs if students attempt to assign Cartesian coordinates to $\vec{a}, \vec{b}, \vec{c}$ and compute the cross products brute-force, which is computationally devastating.

The elegant trap lies in recognizing that the Vector Triple Product $\vec{a} \times (\vec{r} \times \vec{a})$ acts as a projection operator $\vec{r} - (\vec{r}\cdot\vec{a})\vec{a}$ .

Furthermore, expanding the scalar triple product of these projections requires treating the basis vectors dynamically rather than as static coordinates, revealing the linear transformation identity $[\vec{P}_1, \vec{P}_2, \vec{P}_3] = (xBC + yAC + zAB - ABC)[\vec{a},\vec{b},\vec{c}]$ .

Failing to leverage the distributive properties of the scalar triple product leads to an insurmountable algebraic mess.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Projection vs. Coordinate
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free