If the vectors a and b are such that a = 3, b = 2/3 and a is a unit vector, then find the angle between a and b.

MathsCBSE Class 12 · Board exam 2023, set 65/5/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2023: question 24(a)

Set in the CBSE Class 12 board exam of 2023, paper 65/5/1, section B, question 24(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q24(a) · 2 marks · Very short answerVector Algebra

If the vectors $a$ and $b$ are such that $∣ a ∣ = 3$ , $∣ b ∣ = \frac{2}{3}$ and $a \times b$ is a unit vector, then find the angle between $a$ and $b$ .

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

A unit vector has magnitude 1.

Hint 2

Use $∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ$ and substitute the 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 $a$ and $b$ .
Since $a \times b$ is a unit vector, $∣ a \times b ∣ = 1$ .
So $∣ a ∣∣ b ∣ sin θ = 1$ , i.e. $3 \cdot \frac{2}{3} sin θ = 1$ .
1 mark
$sin θ = \frac{1}{2}$ , so $θ = \frac{π}{6}$ (i.e. $3 0^{\circ}$ ).
1 mark

Answer: $θ = \frac{π}{6}$ (30°)

Where the marks are

Uses $∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ = 1$ (1 mark)
Gets $sin θ = \frac{1}{2}$ and $θ = \frac{π}{6}$ (1 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 paper2023 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2023: question 24(a)

Hints

Step-by-step solution

Revise the chapter

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