Vectors Questions, Properties and Step-by-Step Examples

Vectors

  • Vectors are a fundamental concept in mathematics and are used extensively in many fields, including physics, engineering, and computer science.
  • Vectors play a key role in linear algebra, offering a versatile framework for solving systems of linear equations and understanding transformations.

In this article, we will discuss: 

  1. What is a Vector?
  2. We will explore how they are different from scalars, and how to perform vector operations such as addition, subtraction, and multiplication.

Here is one more link to practice a few extra questions: Maths Genie Vectors Questions

What is a Vector?

  • A vector is a mathematical object that has both magnitude and direction.
  • Vectors can be represented geometrically as arrows, where the length of the arrow represents the magnitude of the vector, and the direction of the arrow represents the direction of the vector. For example, consider the following vector:

This vector has a magnitude of 5 and points in the direction of the vector (3, 4). Vectors can have any number of dimensions, but we will focus on two-dimensional vectors in this article.

Vector vs Scalar

  • In mathematics, a scalar is a quantity characterized solely by its magnitude, devoid of any directional component. Time, temperature, and mass serve as examples of scalars.
  • In contrast, vectors have both magnitude and direction. For example, the temperature of a room can be measured using a scalar quantity, but the wind velocity and direction are vector quantities.

Vectors Addition and Subtraction

  • Vector addition and subtraction are performed component-wise.
  • To add or subtract two vectors, we add or subtract their corresponding components.
  • For example, consider the following vectors:

u = (1, 2) v = (3, 4)

The sum of these vectors is:

u + v = (1+3, 2+4) = (4, 6)

The difference between these vectors is:

u – v = (1-3, 2-4) = (-2, -2)

Vectors Multiplication

  • There are two types of vector multiplication: dot product and cross product. However, we will only focus on vector multiplication by a scalar in this article.
  • Vector multiplication by a scalar involves multiplying each component of the vector by the scalar.
  • For example, consider the vector:

v = (3, 4)

Multiplying this vector by the scalar 2 gives:

2v = (23, 24) = (6, 8)

This has the effect of scaling the vector by a factor of 2.

Conclusion

  • In conclusion, vectors are an essential concept in mathematics and have many applications in the real world.
  • Understanding vector operations, including addition, subtraction, and multiplication, is crucial for mastering many mathematical and scientific fields.

Practice Questions: Vectors

Question 1: Given vector v = (2, 3), calculate its magnitude.

Question 2: Given vector u = (-1, 5) and vector v = (3, -2), find the sum u + v.

Question 3: Given vector a = (4, -2) and vector b = (-1, 3), find the difference a – b.

Question 4: Given vector u = (2, 3) and scalar k = 4, find the scalar multiplication k x u.

Question 5: Given vector v = (3, 1), find the unit vector in the same direction as v.

Question 6: Given vector u = (1, 2) and vector v = (4, -1), find the dot product u x v.

Question 7: Given vector u = (2, -3) and vector v = (4, 1), find the magnitude of the vector sum u + v.

Question 8: Given vector u = (3, -1) and vector v = (-2, 5), find the scalar projection of u onto v.

Question 9:

Vectors Questions

OA=5a

OB=3b

M is the midpoint of AB

(a) Find, in terms of a and b, the vector AB

(b) Find, in terms of a and b, the vector AM

(c) Find, in terms of a and b, the vector OM

Question 10: Given vector v = (5, -3), find a vector that is parallel to v but has half its magnitude.

Solutions:

Question 1: Given vector v = (2, 3), calculate its magnitude.

Solution:

The magnitude of vector v = (2, 3) is calculated using the formula:

|v| = √(22 + 32)

= √(4 + 9)

= √13.

Question 2: Given vector u = (-1, 5) and vector v = (3, -2), find the sum u + v.

Solution:

The sum of vectors u = (-1, 5) and v = (3, -2) is calculated by adding their corresponding components:

u + v = (-1 + 3, 5 + (-2))

= (2, 3).

Question 3: Given vector a = (4, -2) and vector b = (-1, 3), find the difference a – b.

Solution:

The difference of vectors a = (4, -2) and b = (-1, 3) is calculated by subtracting their corresponding components:

a – b = (4 – (-1), -2 – 3)

= (5, -5).

Question 4: Given vector u = (2, 3) and scalar k = 4, find the scalar multiplication k x u.

Solution:

The scalar multiplication of scalar k = 4 and vector u = (2, 3) is calculated by multiplying each component of the vector by the scalar:

k x u = (4 x 2, 4 x 3)

= (8, 12).

Question 5: Given vector v = (3, 1), find the unit vector in the same direction as v.

Solution:

The unit vector in the same direction as vector v = (3, 1) is calculated by dividing the vector by its magnitude:

|v| = √(32 + 12)

= √(9 + 1)

= √10.

The unit vector is then v̂ = (3/√10, 1/√10).

Question 6: Given vector u = (1, 2) and vector v = (4, -1), find the dot product u x v.

Solution:

The dot product of vectors u = (1, 2) and v = (4, -1) is calculated by multiplying their corresponding components and summing the results:

u x v = 1 x 4 + 2 x (-1)

= 4 – 2

= 2.

Question 7: Given vector u = (2, -3) and vector v = (4, 1), find the magnitude of the vector sum u + v.

Solution:

The vector sum of u = (2, -3) and v = (4, 1) is calculated by adding their corresponding components:

u + v = (2 + 4, -3 + 1)

= (6, -2).

The magnitude of vector u + v = (6, -2) is calculated using the formula:

|u + v| = √(62 + (-2)2)

= √(36 + 4)

= √40

= 2√10.

Question 8: Given vector u = (3, -1) and vector v = (-2, 5), find the scalar projection of u onto v.

Solution:

The scalar projection of vector u = (3, -1) onto vector v = (-2, 5) is calculated using the formula:

Scalar projection of u onto v = |u| cos(θ),

Here, θ represents the angle formed between vectors u and v.

First, calculate the dot product:

u · v = 3 x (-2) + (-1) x 5 = -6 – 5 = -11.

Then, calculate the magnitude of u:

|u| = √(32 + (-1)2)

= √(9 + 1)

= √10.

Now, substitute these values into the formula:

Scalar projection of u on to

v = (-11) / √10 ≈ -3.48.

Question 9:

Vectors Questions

OA=5a

OB=3b

M is the midpoint of AB

(a) Find, in terms of a and b, the vector AB

(b) Find, in terms of a and b, the vector AM

(c) Find, in terms of a and b, the vector OM

Solution:

(a) -5a + 3b

(b) -5/2 a + 3/2 b

(c) 5/2 a + 3/2 b

Question 10: Given vector v = (5, -3), find a vector that is parallel to v but has half its magnitude.

Solution:

To find a vector parallel to v = (5, -3) with half its magnitude, we can scale the vector by multiplying each component by 0.5:

A vector parallel to v with half its magnitude is

u = (0.5 x 5, 0.5 x (-3))

= (2.5, -1.5).