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Question #41828

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{u} = \hat{i} + 2\hat{j} - \hat{k}$ and $\vec{v} = 2\hat{i} - \hat{j} + \hat{k}$ . A vector $\vec{w} = x\hat{i} + y\hat{j} + z\hat{k}$ is chosen at random such that $x, y, z \in \{-1, 0, 1\}$ . If $P$ is the probability that the volume of the parallelepiped formed by $\vec{u}, \vec{v}$ , and $\vec{w}$ is strictly less than 6, then the value of $P$ is:

Options

A.
$\frac{19}{27}$
B.
$\frac{23}{27}$
C.
$\frac{8}{27}$
D.
$\frac{21}{27}$

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

Complementary Counting|Scalar Triple Product Volume Logic|Dot Product Projection Sign

Why This Question Is Tricky

Common Mistakes

This question tests the application of complementary counting in 3D geometry while embedding a classic 'Dot Product Projection Sign' trap.

Students often calculate the scalar triple product $S = x - 3y - 5z$ and use complementary counting as $1 - P(S \ge 6)$ .

They forget that geometric volume is the absolute value of the scalar triple product, $|S|$ .

By ignoring the negative projection cases ( $S \le -6$ ), they only subtract half of the unfavorable outcomes, leading directly to the trap answer 23/27.

This penalizes superficial formula application over true spatial reasoning.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Complementary Counting
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free