An ant is moving along the vector l_1=i-2j+3k. Few sugar crystals are kept along the vector l_2=3i-2j+k which is inclined at an angle θ with the vector l_1. Then find the angle θ. Also find the scalar projection of l_1 on l_2.

θ=^-157; scalar projection =(10)/(√(14))

MathsCBSE Class 12 · Sample paper, session 2024-25 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2024-25: question 28(a)

From the official CBSE sample question paper for 2024-25, section C, question 28(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28(a) · 3 marks · Short answerCompetency-basedVector Algebra

An ant is moving along the vector $l_{1} = \hat{i} - 2 \hat{j} + 3 \hat{k}$ . Few sugar crystals are kept along the vector $l_{2} = 3 \hat{i} - 2 \hat{j} + \hat{k}$ which is inclined at an angle $θ$ with the vector $l_{1}$ . Then find the angle $θ$ . Also find the scalar projection of $l_{1}$ on $l_{2}$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

The angle between two vectors comes from $a \cdot b = ∣ a ∣∣ b ∣ cos θ$ .

Hint 2

Projection of $a$ on $b$ is $a \cdot \hat{b}$ .

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.

$cos θ = \frac{l _{1} \cdot l _{2}}{∣ l _{1} ∣∣ l _{2} ∣}$ , with $l_{1} \cdot l_{2} = (1) (3) + (- 2) (- 2) + (3) (1) = 10$ .
1 mark
$∣ l_{1} ∣ = 14$ , $∣ l_{2} ∣ = 14$ , so $cos θ = \frac{10}{14} = \frac{5}{7}$ , $θ = cos^{- 1} \frac{5}{7}$ .
½ mark
Scalar projection of $l_{1}$ on $l_{2}$ $= \frac{l _{1} \cdot l _{2}}{∣ l _{2} ∣}$ .
1 mark
$= \frac{10}{14}$ .
½ mark

Answer: $θ = cos^{- 1} \frac{5}{7}$ ; scalar projection $= \frac{10}{14}$

Where the marks are

Uses dot-product formula for angle with $l_{1} \cdot l_{2} = 10$ (1 mark)
$θ = cos^{- 1} (5/7)$ (½ mark)
Formula for scalar projection (1 mark)
Value $10/ 14$ (½ mark)

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Chapter: Vector Algebra (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Vector Algebra, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2024-25: every question, by section.

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CBSE Class 12 Mathematics sample paper 2024-25: question 28(a)

Hints

Step-by-step solution

Revise the chapter

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