Find the shortest distance between the lines : r = (2i - j + 3k) + λ(i - 2j + 3k) r = (i + 4k) + μ(3i - 6j + 9k).

MathsCBSE Class 12 · Board exam 2025, set 65/7/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2025: question 31(b)

Set in the CBSE Class 12 board exam of 2025, paper 65/7/1, section C, question 31(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31(b) · 3 marks · Short answerThree Dimensional Geometry

Find the shortest distance between the lines :

$r = (2 \hat{i} - \hat{j} + 3 \hat{k}) + λ (\hat{i} - 2 \hat{j} + 3 \hat{k})$

$r = (\hat{i} + 4 \hat{k}) + μ (3 \hat{i} - 6 \hat{j} + 9 \hat{k})$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/7/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Compare the two direction vectors. One is a multiple of the other.

Hint 2

For parallel lines, $S D = \frac{∣ b \times ( a _{2} - a _{1} ) ∣}{∣ b ∣}$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

$3 \hat{i} - 6 \hat{j} + 9 \hat{k} = 3 (\hat{i} - 2 \hat{j} + 3 \hat{k})$ , so the lines are parallel with $b = \hat{i} - 2 \hat{j} + 3 \hat{k}$ . Here $a_{1} = 2 \hat{i} - \hat{j} + 3 \hat{k}$ and $a_{2} = \hat{i} + 4 \hat{k}$ , so $a_{2} - a_{1} = - \hat{i} + \hat{j} + \hat{k}$ .
½ mark
$b \times (a_{2} - a_{1}) = \hat{i} 1 - 1 \hat{j} - 2 1 \hat{k} 31 = \hat{i} (- 2 - 3) - \hat{j} (1 + 3) + \hat{k} (1 - 2) = - 5 \hat{i} - 4 \hat{j} - \hat{k}$
1½ marks
$S D = \frac{∣ b \times ( a _{2} - a _{1} ) ∣}{∣ b ∣} = \frac{25 + 16 + 1}{1 + 4 + 9} = \frac{42}{14} = 3$ units
1 mark

Answer: $3$ units

Where the marks are

Recognises the lines are parallel and finds $a_{2} - a_{1}$ (½ mark)
Computes $b \times (a_{2} - a_{1}) = - 5 \hat{i} - 4 \hat{j} - \hat{k}$ (1½ marks)
Shortest distance $= 3$ (1 mark)

Revise the chapter

Chapter: Three Dimensional Geometry (notes, formulas, where marks are lost and every past question)
Topic: Shortest Distance Between Two Lines

Practise this chapterBoard-style questions from Three Dimensional Geometry, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 65/7/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 31(b)

Hints

Step-by-step solution

Revise the chapter

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