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 \vec{a} = 3 \hat{i} + 4\hat{j} .

Magnitude: To find magnitude of a vector a, we use formula ∣∣v∣∣ = \sqrt{a^2+b^2}

|\vec{a}| = \sqrt{(3)^2+(4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5

Unit Vector: \hat{a} = \frac{\vec{a}}{|\vec{a}|} = \frac{3\hat{i}+4{\vec{j}}}{5} = 0.6\hat{i}+0.8\hat{j}

Question 2: Find the Length of the vector \vec{v} = (1, 2, 3).

To find magnitude of a vector v = (a, b, c) we use formula ∣∣v∣∣ = \sqrt{a^2+b^2+c^2}

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

∣v∣∣ = \sqrt{1^2+2^2+3^2}
= √14

= 3.7417

Question 3: Calculate \vec{a} + \vec{b} and \vec{a} - \vec{b} for given vectors.

\vec{a} = 3\hat{i} + 4\hat{j}

\vec{b} = 5\hat{i} - 2\hat{j}

Addition: \vec{a} + \vec{b} = (3 + 5)\hat{i} + (4 - 2)\hat{j} = 8\hat{i} + 2\hat{j}

Subtraction: \vec{a} - \vec{b} = (3 - 5)\hat{i} + (4 - (-2))\hat{j} = -2\hat{i} + 6\hat{j}

Question 4: Calculate \vec{v_1} + \vec{v_2} and \vec{v_1} - \vec{v_2} For the given vectors.

\vec{v_1} = (1, -2, 3)

\vec{v_2} = (-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 2\vec{a} and -3\vec{b} for \vec{a} = 3\hat{i} + 4\hat{j}, \vec{b} = 5\hat{i} - 2\hat{j}

Scaling \vec{a}

\vec{a} = 2(3\hat{i} + 4\hat{j}) = 6\hat{i} + 8\hat{j}

Scaling \vec{b}

-3\vec{b} = -3(5\hat{i} - 2\hat{j}) = -15\hat{i} + 6\hat{j}

Question 6: Find \vec{a} \cdot \vec{b} for \vec{a} = \hat{i} + 2\hat{j}, \vec{b} = 2\hat{i} - \hat{j}

\vec{a} \cdot \vec{b} = (1)(2) + (2)(−1) = 2 − 2 = 0

Question 7: Check if \vec{a} = 2\hat{i} + 3\hat{j}​ and \vec{b} = 4\hat{i} + 6\hat{j}​ are collinear.

Two vectors are collinear if \vec{b} = k\vec{a} for some scalar k.

Since \vec{b} = 2\vec{a}, the vectors are collinear.

Question 8: Find the Angle between vectors \vec{a} = \hat{i} + \hat{j} and \vec{b} = \hat{i} -\hat{j}

The angle between vectors a and b is given by cos(𝛉) = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|{|\vec{b}|}}

First we will find the dot product and magnitudes:

\vec{a} \cdot \vec{b} = 0

|\vec{a}| = \sqrt{1^2+1^2} =\sqrt2\\|\vec{b}| = \sqrt{1^2+(-1)^2}=\sqrt2

Angle:

cos(𝛉) = \frac{0}{\sqrt{2}\cdot\sqrt{2}} = 0

𝛉 = 90°

Practice Questions

Question 1: Find the magnitude and unit vector of \vec{a} = -7\hat{i} + 24\hat{j} .

Question 2: Find the length of the vector \vec{u} = (4, -1, 7)

Question 3: For \vec{a} = 6\hat{i} - 2\hat{j}​ and \vec{b} = -3\hat{i} + 4\hat{j}​, calculate \vec{a} + \vec{b} and \vec{a} - \vec{b}.

Question 4: Calculate \vec{a} and -5\vec{b} for \vec{a} = \hat{i} + 3\hat{j}, \vec{b} = 2\hat{i} - \hat{j} \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 \vec{a} = 5\hat{i} - 10\hat{j}​ and \vec{b} = -15\hat{i} + 30\hat{j}​ are collinear.

Question 7: Compute \vec{a} \cdot \vec{b} for \vec{a} = 2\hat{i} - \hat{j} + \hat{k}, \vec{b} = -\hat{i} + 3\hat{j} - 2\hat{k}.

Question 8: Determine the angle between \vec{a} = 8\hat{i} - 6\hat{j}​ and \vec{b} = 7\hat{i} + 24\hat{j} .

Asnwer Key

1. Magnitude of vector a: 25, Unit Vector: -0.28\hat{i}+0.96\hat{j}
2. Length is √66
3. \vec{a} + \vec{b} = 3\hat{i} + 2\hat{j}, \vec{a} - \vec{b} = 9\hat{i} - 6\hat{j}
4. 4\vec{a} = 4\hat{i} + 12\hat{j}, -5\vec{b} = -10\hat{i} + 5\hat{j}
5. Position Vector from A to B: [4, −8, −5]
6. Collinearity: True
7. Dot Product: -7
8. Angle between vectors: 110.61°