CBSE Class 12 Vector Algebra PYQs Analysis (2021-2025)

Based on detailed analysis of last 5 years' papers. Perfect for 2026 Boards prep!

Key Trends

Weightage Breakdown: 1-2 MCQs/assertion-reason (1 mark), 1-2 short answers (2-3 marks), 1 long answer on scalar/vector triple product or coplanarity (4-5 marks).
Repeated Topics: Scalar & vector products, angle between vectors, projection of vector, section formula in vectors, coplanarity of three vectors, scalar triple product (volume), vector triple product identity, shortest distance between two lines (basic), collinear vectors.
Difficulty: Easy to medium; formula application & algebraic manipulation key. More objective questions in recent years.
Changes: Increased focus on vector triple product & coplanarity in 2023–2025 papers.

Most Important and Repeated Questions

Question Example	Type/Marks	Years Repeated	Notes
If a · b = \|a\|\|b\| cos θ, find angle between vectors a = i + j + k and b = i - j + k (or similar).	Short Answer (2 marks)	2021, 2022, 2023, 2024, 2025	Repeated 5x; cos θ = (a·b)/(\|a\|\|b\|); θ = cos⁻¹(1/3) or similar.
Prove that scalar triple product [a b c] = a · (b × c) represents volume of parallelepiped.	Proof (3 marks)	2021 Term 2, 2022, 2023, 2024	Repeated 4x; Geometric interpretation + algebraic proof.
Show that vectors a, b, c are coplanar if [a b c] = 0 (or check coplanarity of given vectors).	Short Answer (3 marks)	2022, 2023, 2024, 2025	Repeated 4x; Compute scalar triple product = 0 → coplanar.
Assertion: If a · b = 0, then a and b are perpendicular. Reason: Dot product zero implies cos θ = 0 → θ = 90°.	Assertion-Reason (1 mark)	2023, 2024, 2025	Repeated 3x; Both true, reason explains.
Find the projection of vector a on b: proj_b a = (a·b / \|b\|²) b or find unit vector in direction of a + b.	Short Answer (2-3 marks)	2021 Term 1, 2022, 2023, 2025	Repeated 4x; Direct formula application.
Find vector joining points A(1,2,3) and B(4,5,6) or section formula: divide join of A & B in ratio m:n.	Short Answer (2 marks)	2022, 2023, 2024	Repeated 3x; Position vector = (m B + n A)/(m+n).
MCQ: The magnitude of vector a × b is: (a) \|a\|\|b\| sin θ (b) \|a\|\|b\| cos θ (c) \|a\|\|b\| (d) 0	MCQ (1 mark)	2021 Term 1, 2023, 2024	Repeated 3x; Answer (a) \|a\|\|b\| sin θ (area of parallelogram).
Prove vector triple product identity: a × (b × c) = (a·c)b - (a·b)c.	Proof (3 marks)	2023, 2024	Repeated 2x; Component-wise proof or geometric.
Case-based: Given vectors, find if collinear/coplanar, or compute scalar/vector products.	Case-Based (4 marks)	2023, 2025	Repeated 2x; Use dot/cross/scalar triple product.
Find shortest distance between two skew lines r = a1 + λ b1 and r = a2 + μ b2.	Short Answer (3 marks)	2021 Term 2, 2024	Repeated 2x; Distance = \|(a2 - a1) · (b1 × b2)\| / \|b1 × b2\|.

Best Format for Preparation and Answering

Study Format: Memorize all vector formulas (dot, cross, triple products). Practice coplanarity & projection daily. Make table for section formula & angle between vectors.
Answer Format: Show all steps: compute dot/cross product, magnitude, then conclude. For proofs: Use definition or component method. Always write unit vector as vector / magnitude.
Tips: Scalar triple product (coplanarity/volume) & projection are highest repeat & marks. Practice 10 PYQs daily. Time: 8–10 mins for 4–5 mark questions. Very scoring if calculations are accurate.

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