The diagonals of a parallelogram are given by a=2i-j+k and b=i+3j-k. Find the area of the parallelogram.

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

CBSE Class 12 Mathematics 2025: question 23

Set in the CBSE Class 12 board exam of 2025, paper 65/1/1, section B, question 23, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23 · 2 marks · Very short answerVector Algebra

The diagonals of a parallelogram are given by $a = 2 \hat{i} - \hat{j} + \hat{k}$ and $b = \hat{i} + 3 \hat{j} - \hat{k}$ . Find the area of the parallelogram.

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 given vectors are the diagonals, not the sides. The area formula is different for diagonals.

Hint 2

For diagonals $d_{1}, d_{2}$ , area $= \frac{1}{2} ∣ d_{1} \times d_{2} ∣$ . Work out the determinant.

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.

Find the cross product of the diagonals: $a \times b = \hat{i} 21 \hat{j} - 1 3 \hat{k} 1 - 1 = \hat{i} (1 - 3) - \hat{j} (- 2 - 1) + \hat{k} (6 + 1) = - 2 \hat{i} + 3 \hat{j} + 7 \hat{k}$ .
1 mark
When $a$ and $b$ are the diagonals, area of parallelogram $= \frac{1}{2} ∣ a \times b ∣ = \frac{1}{2} (- 2)^{2} + 3^{2} + 7^{2} = \frac{1}{2} 62 = \frac{62}{2}$ sq units.
1 mark

Answer: Area $= \frac{62}{2}$ sq units

Where the marks are

Correct $a \times b = - 2 \hat{i} + 3 \hat{j} + 7 \hat{k}$ (1 mark)
Uses area $= \frac{1}{2} ∣ a \times b ∣$ to get $\frac{62}{2}$ (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 paper2025 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 23

Hints

Step-by-step solution

Revise the chapter

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