Vector Algebra — Chapter Test | Class 12 IMO

Q1

Area of the parallelogram formed by vectors 2i and 3j is:

|2i×3j| = 6.

Q2

The value of k×i is:

k×i = j.

Q3

Area of the triangle formed by i and j is:

Triangle area = 1/2 × |i×j|.

Q4

If the vectors a = 2î + λĵ + k̂ and b = î − 2ĵ + 3k̂ are perpendicular to each other, find the value of λ.

For perpendicular vectors, a.b = 0. So, (2)(1) + (λ)(−2) + (1)(3) = 0 → 2 − 2λ + 3 = 0 → 5 = 2λ → λ = 5/2.

Q5

Two oxen are pulling a cart in an Indian farm. One pulls with a force F₁ = 10î + 5ĵ N, and the other pulls with F₂ = 5î − 5ĵ N. What is the resultant force on the cart?

Resultant Force F = F₁ + F₂ = (10î + 5ĵ) + (5î − 5ĵ) = 15î N. The forces in the j direction cancel out.

Q6

If a × b = a × c and a ≠ 0, then which of the following is definitely true?

a × b − a × c = 0 → a × (b − c) = 0. This means that either b − c = 0 (so b = c), or vector a is parallel to vector (b − c). Thus, b=c is not 'definitely' true, but a being parallel to (b−c) encompasses the general relationship (including the zero vector case trivially).

Q7

If a = 2i + 3j + k and b = i − j + 2k, then a·b equals:

2−3+2=1.

Q8

If two vectors are collinear, which of the following statements is ALWAYS false?

Collinear vectors are parallel but can have opposite directions (anti-parallel). Thus, they don't must have the same direction.

Q9

Rani applies a force F = 2î + 3ĵ on a trolley, which moves along the line r = î + ĵ. The component of the force driving the trolley forward (projection) is:

Projection of F on r = (F.r) / |r| = (2(1) + 3(1)) / √(1²+1²) = 5 / √2.

Q10

A shopkeeper moves ₹500 worth of goods 4 units east and 3 units north represented by vectors. The resultant displacement magnitude is:

√(4²+3²)=5.

Q11

For any two vectors a and b, what is the geometric interpretation of |a × b|?

The magnitude of the cross product of two vectors represents the area of a parallelogram whose adjacent sides are given by those vectors.

Q12

Which of the following is equal to the scalar triple product [b c a]?

Cyclic permutations of the vectors in a scalar triple product leave its value unchanged. Hence, [b c a] = [a b c].

Q13

Vectors a and b satisfy |a×b| = |a||b|. Then angle between them is:

sinθ = 1 gives θ = 90°.

Q14

The angle between vectors with positive dot product is:

cosθ > 0 implies θ is acute.

Q15

If a = 2i, b = 3j and c = 4k, volume of the parallelepiped is:

|[a,b,c]| = 2×3×4 = 24.