Vector Algebra — Class 12 MCQs with Answers

Vector Algebra — important questions & MCQs with answers (Class 12 Mathematics)

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   Magnitude of (3,4):

   

   A5B7C25D4
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   Answer: 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).

2. Q2 of 6Easy

   i · j =

   A1B0CkD−1
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   Answer: 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.

3. Q3 of 6Medium

   Magnitude of 3î + 4ĵ:

   

   A5B7C12D25
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   Answer: 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.

4. Q4 of 6Medium

   Assertion (A): The vectors a⃗ = 2î + ĵ + 3k̂ and b⃗ = î + ĵ − k̂ are perpendicular to each other. Reason (R): Two non-zero vectors are perpendicular if and only if their scalar product is zero.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true and R is the correct explanation of A

   Explanation: A is true: a⃗ · b⃗ = 2(1) + 1(1) + 3(−1) = 0 and both vectors are non-zero. R is true: it is Observation 2 for the scalar product. A follows directly by applying R to the zero scalar product, so R explains A.

5. Q5 of 6Hard

   a⃗ × (b⃗ + c⃗) = ?

   ARandomBa⃗ × b⃗Cb⃗ × a⃗ + c⃗ × a⃗Da⃗ × b⃗ + a⃗ × c⃗
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   Answer: a⃗ × b⃗ + a⃗ × c⃗

   Explanation: Cross product distributes over addition just like ordinary multiplication: a⃗×(b⃗+c⃗) = a⃗×b⃗+a⃗×c⃗. Swapping to b⃗×a⃗+c⃗×a⃗ would flip both signs since order matters.

6. Q6 of 6Hard

   Assertion (A): If a⃗ and b⃗ are unit vectors and a⃗ + b⃗ is also a unit vector, then the angle between a⃗ and b⃗ is 2π/3. Reason (R): For any two vectors a⃗ and b⃗, |a⃗ + b⃗|² = |a⃗|² + |b⃗|² + 2 a⃗ · b⃗.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true and R is the correct explanation of A

   Explanation: R is true: expand (a⃗ + b⃗) · (a⃗ + b⃗) using distributivity and commutativity. Using R, 1 = 1 + 1 + 2 cos θ, so cos θ = −1/2 and θ = 2π/3; A is true. A is obtained exactly by applying R, so R explains A.

Vector Algebra notes: key concepts (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.

Position vector & section formula

P(x, y, z) has position vector x î + y ĵ + z k̂; the point dividing AB in ratio m:n internally is (m b⃗ + n a⃗)/(m + n).

Dot (scalar) product

a⃗·b⃗ = |a⃗||b⃗|cosθ = a₁b₁ + a₂b₂ + a₃b₃; it is commutative and returns a scalar.

Cross (vector) product

a⃗×b⃗ has magnitude |a⃗||b⃗|sinθ, direction perpendicular to both by the right-hand rule; it is anti-commutative.

Perpendicular & parallel tests

Non-zero vectors are perpendicular iff a⃗·b⃗ = 0 and parallel iff a⃗×b⃗ = 0⃗.

Projection of a vector

The scalar projection of a⃗ on b⃗ is (a⃗·b⃗)/|b⃗|; it measures how much of a⃗ lies along b⃗.

Angle between vectors

cosθ = (a⃗·b⃗)/(|a⃗||b⃗|) gives the angle; the cross product gives sinθ instead.

Area via cross product

|a⃗×b⃗| is the area of the parallelogram on a⃗ and b⃗; the triangle is ½|a⃗×b⃗|, and diagonals give ½|d⃗₁×d⃗₂|.

Scalar triple product

[a⃗ b⃗ c⃗] = a⃗·(b⃗×c⃗) equals the signed volume of the parallelepiped; the vectors are coplanar iff it is 0.