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Question #40875

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{v}_1 = 2\hat{i} + 2\hat{j} + \hat{k}$ , $\vec{v}_2 = 4\hat{i} + 7\hat{j} - 4\hat{k}$ , and $\vec{v}_3 = 8\hat{i} - \hat{j} + 4\hat{k}$ . Let $\vec{b}_1$ be a vector along the acute angle bisector of $\vec{v}_1$ and $\vec{v}_2$ , and $\vec{b}_2$ be a vector along the acute angle bisector of $\vec{v}_2$ and $\vec{v}_3$ . If the vector $\vec{u} = 14\hat{i} + \lambda\hat{j} + 2\hat{k}$ is coplanar with $\vec{b}_1$ and $\vec{b}_2$ , then the value of $\lambda$ is:

Options

A.
3
B.
-2
C.
-9
D.
-54

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

Collinearity and Coplanarity|Geometric Interpretation of Cross Product|Angle Bisector Direction

Why This Question Is Tricky

Common Mistakes

This question tests the geometric interpretation of the cross product and the precise definition of angle bisectors in 3D space.

The primary trap lies in the dual-nature of angle bisectors and the necessity of normalization.

Students frequently make the mistake of directly adding the given direction vectors ( $\vec{v}_1 + \vec{v}_2$ ) without normalizing them first, which yields an incorrect bisector direction and leads to the phantom answer $\lambda = 3$ .

Furthermore, the problem requires verifying whether the sum or difference of the unit vectors corresponds to the acute angle bisector by checking the sign of their dot product.

Assuming the difference of unit vectors (obtuse bisector) leads to another trap answer, $\lambda = -54$ .

Quick Facts About This Question

Question Information

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