Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors a=3i+j+2k and b=2i-2j+4k. Determine the angle formed between the kite strings. Assume there is no slack in the strings.

θ = ^-1 ((3)/(√(21)) ), which equals ^-1 ((12)/(√(336)) )

MathsCBSE Class 12 · Board exam 2025, set 65/1/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2025: question 25(a)

Set in the CBSE Class 12 board exam of 2025, paper 65/1/1, section B, question 25(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q25(a) · 2 marks · Very short answerCompetency-basedVector Algebra

Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors $a = 3 \hat{i} + \hat{j} + 2 \hat{k}$ and $b = 2 \hat{i} - 2 \hat{j} + 4 \hat{k}$ . Determine the angle formed between the kite strings. Assume there is no slack in the strings.

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

Hints

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

Hint 1

The angle between two vectors comes from their dot product.

Hint 2

Use $cos θ = \frac{a \cdot b}{∣ a ∣∣ b ∣}$ , then find the dot product and both magnitudes.

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.

Let $θ$ be the angle between the strings, so $cos θ = \frac{a \cdot b}{∣ a ∣∣ b ∣}$ . Here $a \cdot b = 6 - 2 + 8 = 12$ , $∣ a ∣ = 9 + 1 + 4 = 14$ and $∣ b ∣ = 4 + 4 + 16 = 24$ . So $cos θ = \frac{12}{336} = \frac{12}{4 21} = \frac{3}{21}$ .
1½ marks
Therefore $θ = cos^{- 1} (\frac{3}{21})$ .
½ mark

Answer: $θ = cos^{- 1} (\frac{3}{21})$ , which equals $cos^{- 1} (\frac{12}{336})$

Where the marks are

Applies the dot-product formula and gets $cos θ = \frac{12}{336} = \frac{3}{21}$ (1½ marks)
States $θ = cos^{- 1} \frac{3}{21}$ (½ mark)

Revise the chapter

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 paper2025 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 25(a)

Hints

Step-by-step solution

Revise the chapter

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