Find the distance between the lines : r = (i + 2j - 4k) + λ(2i + 3j + 6k); r = (3i + 3j - 5k) + μ(4i + 6j + 12k)

MathsCBSE Class 12 · Board exam 2023, set 65/5/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2023: question 31

Set in the CBSE Class 12 board exam of 2023, paper 65/5/1, section C, question 31, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31 · 3 marks · Short answerThree Dimensional Geometry

Find the distance between the lines :

$r = (\hat{i} + 2 \hat{j} - 4 \hat{k}) + λ (2 \hat{i} + 3 \hat{j} + 6 \hat{k})$ ;

$r = (3 \hat{i} + 3 \hat{j} - 5 \hat{k}) + μ (4 \hat{i} + 6 \hat{j} + 12 \hat{k})$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/5/1.

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 use $d = \frac{∣ ( a _{2} - a _{1} ) \times b ∣}{∣ b ∣}$ .

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.

Here $a_{1} = \hat{i} + 2 \hat{j} - 4 \hat{k}$ and $a_{2} = 3 \hat{i} + 3 \hat{j} - 5 \hat{k}$ .
The directions are $2 \hat{i} + 3 \hat{j} + 6 \hat{k}$ and $4 \hat{i} + 6 \hat{j} + 12 \hat{k} = 2 (2 \hat{i} + 3 \hat{j} + 6 \hat{k})$ .
So the lines are parallel, with $b = 2 \hat{i} + 3 \hat{j} + 6 \hat{k}$ .
½ mark
$a_{2} - a_{1} = 2 \hat{i} + \hat{j} - \hat{k}$
½ mark
$(a_{2} - a_{1}) \times b = \hat{i} 22 \hat{j} 13 \hat{k} - 1 6$
$= \hat{i} (6 + 3) - \hat{j} (12 + 2) + \hat{k} (6 - 2) = 9 \hat{i} - 14 \hat{j} + 4 \hat{k}$
1 mark
Distance $d = \frac{∣ ( a _{2} - a _{1} ) \times b ∣}{∣ b ∣} = \frac{81 + 196 + 16}{4 + 9 + 36} = \frac{293}{7}$ units
1 mark

Answer: $\frac{293}{7}$ units

Where the marks are

Identifies that the lines are parallel (½ mark)
Finds $a_{2} - a_{1}$ (½ mark)
Correct cross product $9 \hat{i} - 14 \hat{j} + 4 \hat{k}$ (1 mark)
Applies the parallel-lines distance formula to get $\frac{293}{7}$ (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 paper2023 board paper, set 65/5/1: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics 2023: question 31

Hints

Step-by-step solution

Revise the chapter

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