Vector Algebra — Class 12 MCQs with Answers
Class 12 CBSE Mathematics · Chapter 10
90 practice questions · 30 Easy · 30 Medium · 30 Hard · Updated
Practise the most important Class 12 CBSE Mathematics questions from Chapter 10, "Vector Algebra". You get 9 timed quizzes made from 90 NCERT-based MCQs, with answers and explanations. The questions are split into 30 Easy, 30 Medium and 30 Hard. Warm up on the basics, then move on to the exam-level questions that set top scorers in CBSE & Maharashtra HSC Board exams, JEE Main, MHT-CET and JEE Advanced apart.
To score well in "Vector Algebra", focus on fast problem-solving, formula recall and step-by-step working. Each MCQ here is timed and uses exam-style marking (+4 correct, −1 wrong, 0 skipped). This trains you to stay accurate under time pressure, as real papers need. Every question has a short explanation, so a wrong answer becomes a quick lesson. It is the fastest way to fix gaps before a test.
Use this chapter for focused revision. Start with the Easy set to check your basics on Vector Algebra, then move to Medium and Hard to practise applying them. Your accuracy, streaks and XP save automatically. This chapter also adds to your overall Class 12 Mathematics mastery score. 14 sample questions are solved in full below, with the answer and a worked explanation. Sign in free to start practising.
Key concepts: Vector Algebra (Class 12 Mathematics)
Vectors carry both magnitude and direction. The chapter covers addition, scaling, magnitude and unit vectors, the section formula, then the two products: the dot product (a scalar, testing perpendicularity and giving projections) and the cross product (a vector, giving areas), plus the triple product for coplanarity.
- Scalar vs vector
- A scalar has only magnitude (mass, speed); a vector has magnitude and direction (displacement, force) and obeys vector addition.
- Magnitude of a vector
- For v⃗ = a î + b ĵ + c k̂, |v⃗| = √(a² + b² + c²); the zero vector has magnitude 0 and no fixed direction.
- Unit vector
- A vector of magnitude 1; v̂ = v⃗/|v⃗| gives the direction of v⃗. î, ĵ, k̂ are the mutually perpendicular unit vectors.
- Vector addition & scaling
- Added by the triangle/parallelogram law and componentwise; k v⃗ scales length by |k|, reversing direction if k < 0.
Vector Algebra — important questions & MCQs with answers (Class 12 Mathematics)
14 solved questions from this chapter's difficulty levels, each with its answer and explanation. The other 76 are timed and scored when you sign in.
- Q1Easy
Magnitude of (3,4):
A.5✓ CorrectB.7C.25D.4Answer: A. 5
Explanation: Magnitude uses √(x²+y²): √(3²+4²) = √25 = 5. Squaring first then adding avoids the trap of picking 7 (adding 3 and 4) or 25 (forgetting the square root).
- Q2Easy
i · j =
A.1B.0✓ CorrectC.kD.−1Answer: B. 0
Explanation: î and ĵ are perpendicular unit vectors, so î·ĵ = |î||ĵ|cos90° = 1·1·0 = 0. Don't confuse this scalar dot product with the vector cross product î×ĵ = k̂.
- Q3Easy
A vector has:
A.NeitherB.Magnitude onlyC.Direction onlyD.Magnitude and direction✓ CorrectAnswer: D. Magnitude and direction
Explanation: A vector carries both magnitude (size) and direction — that's what separates it from a scalar, which has size alone. Speed is a scalar; velocity, with a direction attached, is a vector.
- Q4Easy
Dot product a⃗ · b⃗ =
A.|a⃗| |b⃗| cos θ✓ CorrectB.|a⃗| |b⃗| sin θC.a⃗ + b⃗D.|a⃗| × |b⃗|Answer: A. |a⃗| |b⃗| cos θ
Explanation: By definition, a⃗·b⃗ = |a⃗||b⃗|cosθ — a scalar. The sinθ version belongs to the cross product's magnitude, and a⃗+b⃗ is a different operation entirely, vector addition.
- Q5Easy
Cross product a⃗ × b⃗ has magnitude:
A.|a⃗| + |b⃗|B.|a⃗| |b⃗| cos θC.|a⃗| |b⃗| sin θ✓ CorrectD.Always 1Answer: C. |a⃗| |b⃗| sin θ
Explanation: The cross product's magnitude is |a⃗×b⃗| = |a⃗||b⃗|sinθ — note sinθ, not cosθ, which belongs to the dot product a⃗·b⃗ instead.
- Q6Easy
A scalar has:
A.Magnitude only✓ CorrectB.Both magnitude and directionC.Direction onlyD.NeitherAnswer: A. Magnitude only
Explanation: A scalar is fully described by magnitude alone, with no direction — mass, temperature, speed. Don't confuse it with a vector, which needs a direction too.
- Q7Medium
a · b = |a||b| ___
A.sin θB.cos θ✓ CorrectC.tan θD.1Answer: B. cos θ
Explanation: The dot product is defined as a⃗·b⃗ = |a⃗||b⃗|cosθ, where θ is the angle between them. sinθ belongs to the cross product's magnitude, not the dot product — that's the trap.
- Q8Medium
Magnitude of 3î + 4ĵ:
A.5✓ CorrectB.7C.12D.25Answer: A. 5
Explanation: Magnitude of 3î+4ĵ is √(3²+4²) = √25 = 5, the same 3-4-5 right triangle as before. Adding the components (3+4=7) instead of squaring is the common slip.
- Q9Medium
a⃗ · b⃗ for a⃗ = 2î + 3ĵ and b⃗ = î − 4ĵ:
A.6B.10C.2D.−10✓ CorrectAnswer: D. −10
Explanation: a⃗·b⃗ = (2)(1) + (3)(−4) = 2 − 12 = −10 — sum the products of matching components. Dropping the negative sign on the second term gives the wrong answer of 10.
- Q10Medium
î × ĵ + ĵ × k̂ + k̂ × î =
A.0B.î + ĵ + k̂✓ CorrectC.2îD.3î + 3ĵAnswer: B. î + ĵ + k̂
Explanation: Each term is one of the standard cyclic products: î×ĵ=k̂, ĵ×k̂=î, k̂×î=ĵ, so the sum is k̂+î+ĵ = î+ĵ+k̂ — not scalar 0, since this is a sum of vectors, not a dot product.
- Q11Medium
Angle θ between a⃗ and b⃗ satisfies:
A.cos θ = a⃗·b⃗/(|a⃗| |b⃗|)✓ CorrectB.sin θ = a⃗·b⃗C.tan θ = a⃗ × b⃗D.cos θ = a⃗ + b⃗Answer: A. cos θ = a⃗·b⃗/(|a⃗| |b⃗|)
Explanation: cosθ = a⃗·b⃗/(|a⃗||b⃗|) comes straight from the dot product definition rearranged for θ. The sinθ and tanθ options mix up the dot product formula with the cross product one.
- Q12Hard
Three vectors are coplanar iff their scalar triple product:
A.= 0✓ CorrectB.≠ 0C.> 0D.< 0Answer: A. = 0
Explanation: Coplanar vectors span zero volume, and the scalar triple product [a⃗,b⃗,c⃗] measures exactly that volume. So coplanar means [a⃗,b⃗,c⃗] = 0, not a nonzero value.
- Q13Hard
|a × b| equals area of:
A.Rectangle onlyB.TriangleC.Parallelogram with sides a and b✓ CorrectD.CubeAnswer: C. Parallelogram with sides a and b
Explanation: The magnitude |a⃗×b⃗| is exactly the area of the parallelogram with sides a⃗ and b⃗. Halve it, ½|a⃗×b⃗|, to get the triangle area — don't stop at the parallelogram number.
- Q14Hard
a⃗ · b⃗ = b⃗ · a⃗ shows that dot product is:
A.Anti-commutativeB.Commutative✓ CorrectC.Distributive onlyD.Non-commutativeAnswer: B. Commutative
Explanation: a⃗·b⃗ = b⃗·a⃗ means swapping the order doesn't change the dot product's value — that's commutativity. The cross product lacks this property: a⃗×b⃗ = −(b⃗×a⃗) instead.
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Start this chapter free →Vector Algebra — FAQs
What are the key concepts in Class 12 Mathematics Vector Algebra?+
Vectors carry both magnitude and direction. The chapter covers addition, scaling, magnitude and unit vectors, the section formula, then the two products: the dot product (a scalar, testing perpendicularity and giving projections) and the cross product (a vector, giving areas), plus the triple product for coplanarity. Key ideas include Scalar vs vector, Magnitude of a vector, Unit vector, Vector addition & scaling.
What does Class 12 Mathematics Chapter 10 (Vector Algebra) cover on XamBaaz?+
It has 90 NCERT-based MCQs on "Vector Algebra": 30 Easy, 30 Medium and 30 Hard. Together they make 9 timed quizzes, and you never get the same set twice. Every question has an instant explanation. They help you prepare for CBSE & Maharashtra HSC Board exams, JEE Main, MHT-CET and JEE Advanced.
Are these "Vector Algebra" questions free to practise?+
Yes. Sign in with Google to practise "Vector Algebra" free. Full unlimited access is ₹999/year on a launch offer until 1 December 2026. No chapter is charged separately.
How should I revise "Vector Algebra" for the exam?+
Start with the Easy quiz to check your basics, then try Medium and Hard to practise applying them. There are 9 timed quizzes on this chapter, so you can come back for a fresh set instead of one you have seen. Read each explanation, retry the questions you miss, and track your accuracy until it stays high.
Are these "Vector Algebra" MCQs available with answers?+
Yes. 14 sample questions are shown here in full, each with the correct option and a step-by-step "Why" explanation. Sign in free with Google to start practising, with instant scoring.
Is there negative marking in the "Vector Algebra" quizzes?+
Yes. The timed quizzes use exam-style marking: +4 for a right answer, −1 for a wrong one and 0 for a skip, the same negative marking as JEE Main and JEE Advanced. MHT-CET and CBSE board papers have no negative marking. Our mocks for those are scored their way.
What are the important questions from Vector Algebra (Class 12 Mathematics)?+
The questions that matter most test Scalar vs vector, Magnitude of a vector, Unit vector, Vector addition & scaling. This page shows 14 solved important MCQs with answers and explanations; all 90 questions on the chapter are available as timed quizzes once you sign in.
Is there an online quiz for Vector Algebra?+
Yes — Class 12 Mathematics Vector Algebra has timed online quizzes at Easy, Medium and Hard levels, with instant scoring and a worked explanation on every question. The first quiz on the chapter is free.