Board 2024 Class 12 Maths Q20 Solution: No line is perpendicular to all three axes

The question

Select the correct answer from the codes (A), (B), (C) and (D) as given below.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).

(C) Assertion (A) is true, but Reason (R) is false.

(D) Assertion (A) is false, but Reason (R) is true.

Assertion (A) : A line in space cannot be drawn perpendicular to x, y and z axes simultaneously.

Reason (R) : For any line making angles, $\alpha$ ⁠, $\beta$ ⁠, $\gamma$ with the positive directions of x, y and z axes respectively, $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$ ⁠.

Exam answer

Reason (R) is true: $\cos\alpha$ ⁠, $\cos\beta$ ⁠, $\cos\gamma$ are the direction cosines $l, m, n$ of the line, and $l^2 + m^2 + n^2 = 1$ ⁠.

Why?

The unit vector along the line is

\hat r = l\,\hat i + m\,\hat j + n\,\hat k

Its components are exactly $\cos\alpha$ ⁠, $\cos\beta$ ⁠, $\cos\gamma$ ⁠, and its length is $1$ ⁠, so

l^2 + m^2 + n^2 = |\hat r|^2 = 1

If a line were perpendicular to all three axes, then $\alpha = \beta = \gamma = \frac{\pi}{2}$ ⁠, and

\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 0 \ne 1

Why?

Perpendicular to an axis means the angle with it is $90^\circ$ ⁠, and $\cos 90^\circ = 0$ ⁠. So all three squares are $0$ ⁠, and their sum is $0$ ⁠, not $1$ ⁠.

\begin{aligned} &\cos^2\tfrac{\pi}{2} + \cos^2\tfrac{\pi}{2} + \cos^2\tfrac{\pi}{2} \\ &= (0)^2 + (0)^2 + (0)^2 \\ &= 0 + 0 + 0 = 0 \end{aligned}

This is impossible by Reason (R). So Assertion (A) is true, and Reason (R) explains it.

Why?

We proved the Assertion by contradiction, and the contradiction $0 = 1$ comes straight from the identity in the Reason. So the Reason is the correct explanation, and the code is (A), not (B).

Answer: (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

Where this comes from

Chapter 11, Three Dimensional Geometry: Direction cosines: l² + m² + n² = 1.

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Q20: No line is perpendicular to all three axes