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Question #41664

Difficulty: 8/10 | 🟢 Free

The Question

Three planes $P_1, P_2,$ and $P_3$ intersect in a common line $L$ . Let $\hat{n}_1, \hat{n}_2,$ and $\hat{n}_3$ be the unit normal vectors to $P_1, P_2,$ and $P_3$ respectively, such that their $\hat{i}$ components are strictly positive. It is given that $2\hat{n}_1 + \hat{n}_2 = \sqrt{5}\hat{n}_3$ . The position vectors of the foot of the perpendiculars from the origin to $P_1$ and $P_3$ are $\vec{v}_1 = \frac{1}{3}(\hat{i} + 2\hat{j} + 2\hat{k})$ and $\vec{v}_3 = \frac{1}{15}(4\hat{i} + 5\hat{j} + 2\hat{k})$ respectively. If $S$ is the sum of the intercepts made by the plane $P_2$ on the coordinate axes, then the value of $S$ is:

Options

A.
$-3$
B.
$3$
C.
$9$
D.
$-9$

Check Your Answer

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

Variable plane intercepts|Linear Independence vs Dependence|Dot Product Projection Sign|Collinearity vs Coplanarity

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the vector equation of a plane and the geometric interpretation of the dot product projection.

The primary trap lies in the sign of the perpendicular distance $d_2$ from the origin.

Students often incorrectly assume that the distance $d$ in $\vec{r} \cdot \hat{n} = d$ must be strictly positive, taking the absolute value $|d_2| = 1$ .

This 'Dot Product Projection Sign' error flips the orientation of the plane, resulting in inverted intercepts and a sum of $3$ instead of $-3$ .

Additionally, the problem requires recognizing that for planes to be concurrent in a line, the linear dependence of their coplanar normal vectors must perfectly mirror the linear combination of their scalar distances, penalizing those who confuse collinearity with coplanarity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Variable plane intercepts
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free