Vector Algebra Practice Questions (Easy)

Vector Algebra is a branch of mathematics that deals with vectors and their operations. A vector is a quantity that has both magnitude and direction, and vector algebra provides the tools to perform calculations and solve problems involving vectors.

Question 1: Find the magnitude and unit vector of .

Magnitude: To find magnitude of a vector a, we use formula ∣∣v∣∣ =

Unit Vector:

Question 2: Find the Length of the vector = (1, 2, 3).

To find magnitude of a vector v = (a, b, c) we use formula ∣∣v∣∣ =

In the above vector a = 1, b = 2 and c = 3 so:

∣v∣∣ =
= √14

= 3.7417

Question 3: Calculate and for given vectors.

Addition:

Subtraction:

Question 4: Calculate and For the given vectors.

= (1, -2, 3)

= (-4, 5, -6)

To find the sum v of vectors v₁ ​= (a₁​, b₁​, c₁​) and v₂ ​= (a₂​, b_2​, c₂​) we use formula: v₁ + v₂ = (a₁ ​+ a₂​, b₁ + b₂​, c₁​ + c₂​)

a₁​ = 1, b₁ ​= −2, c₁​ = 3, a₂ ​= −4 and b₂ ​= 5, c₂ ​= −6, therefore:

v ​= (a₁ ​+ a₂​, b_1​ + b₂​, c₁ ​+ c₂​)
= (1 − 4, −2 + 5, 3 − 6)
= (−3, 3, −3)

To find the difference v of vectors v1​=(a1​,b1​,c1​) and v2​=(a2​,b2​,c2​) we use formula: v₁ - v₂ = (a₁ ​- a₂​, b₁ - b₂​, c₁​ + c₂​)

v = (a₁ ​− a₂​, b₁ ​− b₂​, c₁ ​− c₂​)
= (1 + 4, −2 −5, 3 + 6)
= (5, −7, 9)​

Question 5: Calculate and for ,

Scaling

Scaling

Question 6: Find for ,

= (1)(2) + (2)(−1) = 2 − 2 = 0

Question 7: Check if ​ and ​ are collinear.

Two vectors are collinear if for some scalar k.

Since , the vectors are collinear.

Question 8: Find the Angle between vectors and

The angle between vectors a and b is given by cos(𝛉) =

First we will find the dot product and magnitudes:

Angle:

cos(𝛉) =

𝛉 = 90°

Practice Questions

Question 1: Find the magnitude and unit vector of .

Question 2: Find the length of the vector = (4, -1, 7)

Question 3: For ​ and ​, calculate and .

Question 4: Calculate and for \vec{b} = 2\hat{i} - \hat{j}.

Question 5: Find the vector joining A(2, 5, 7) to B(6, −3, 2).

Question 6: Check if the vectors ​ and ​ are collinear.

Question 7: Compute for

Question 8: Determine the angle between ​ and .

Asnwer Key

1. Magnitude of vector a: 25, Unit Vector:
2. Length is √66
3. ,
4. Position Vector from A to B: [4, −8, −5]
5. Collinearity: True
6. Dot Product: -7
7. Angle between vectors: 110.61°