Vectors: Definition, Operations, and Applications in Math

Question 1

A force of 20 N is applied to an object at an angle of 30 degrees above the horizontal. Calculate the horizontal component of this force.

Accepted Answer

20 N * cos 30°

Answer

20 N * sin 30°

Answer

20 N

Answer

0 N

Question 2

Which of the following operations is not applicable to vectors?

Accepted Answer

Division by a vector

Answer

Addition

Answer

Subtraction

Answer

Multiplication by a scalar

Question 3

Find the unit vector in the direction of the vector (6, 8).

Accepted Answer

(0.6, 0.8)

Answer

(3, 4)

Answer

(6, 8)

Answer

(1, 1)

Question 4

Given vectors A = (2, 3) and B = (-1, 4), determine the angle between A and B.

Accepted Answer

30.96°

Answer

45°

Answer

60°

Answer

90°

Question 5

Which of the following is a defining property of a vector?

Accepted Answer

Has both magnitude and direction

Answer

Has only magnitude, not direction

Answer

Has only direction, not magnitude

Answer

Neither magnitude nor direction

Question 6

Calculate the resultant force acting on an object with mass 'm' moving with velocity 'v'.

Accepted Answer

mv

Answer

mv^2

Answer

½mv^2

Answer

m/v

Question 7

Identify the position vector of a point P(x, y) relative to the origin.

Accepted Answer

(x, y)

Answer

(y, x)

Answer

x + y

Answer

x - y

Question 8

What is the cross product of two parallel vectors?

Accepted Answer

A zero vector

Answer

A vector perpendicular to both vectors

Answer

Equal to the magnitude of the vectors

Answer

Equal to the negative of the magnitude of the vectors

Question 9

Which of the following is **NOT** a vector quantity?

Accepted Answer

Temperature

Answer

Force

Answer

Velocity

Answer

Displacement

Question 10

A force of 10 N is applied at an angle of 30 degrees to the horizontal. What is the horizontal component of the force?

Accepted Answer

8.66 N

Answer

5 N

Answer

10 N

Answer

17.32 N

Question 11

A force of 200 Newtons is applied to an object at an angle of 60 degrees below the horizontal. What is the vertical component of this force?

Accepted Answer

173.2 Newtons

Answer

100 Newtons

Answer

200 Newtons

Answer

346.4 Newtons

Question 12

Which operation is not applicable to vectors?

Accepted Answer

Multiplication

Answer

Subtraction

Answer

Addition

Answer

Dot product

Question 13

What is the angle between the vectors a = i + j and b = -i + j?

Accepted Answer

90 degrees

Answer

180 degrees

Answer

45 degrees

Answer

135 degrees

Question 14

What is the acceleration of an object moving with a non-zero, constant velocity?

Accepted Answer

0 m/s^2

Answer

10 m/s^2

Answer

15 m/s^2

Answer

9.8 m/s^2

Question 15

Which of the following accurately represents a vector in two dimensions?

Accepted Answer

⟨a, b⟩

Answer

a + b

Answer

(a, b)

Answer

a * b

Question 16

Determine the magnitude of the vector v = ⟨3, 4⟩.

Accepted Answer

5

Answer

12

Answer

2

Answer

7

Question 17

Which operation is NOT applicable to vectors?

Accepted Answer

Division

Answer

Addition

Answer

Dot product

Answer

Subtraction

Question 18

A force of 10 newtons acts on an object at an angle of 30° above the horizontal. Calculate the horizontal component of the force.

Accepted Answer

8.66 newtons

Answer

5 newtons

Answer

10 newtons

Answer

17.32 newtons

Question 19

Which of the following is NOT a property of a vector?

Accepted Answer

It is a scalar quantity.

Answer

It has magnitude.

Answer

It has direction.

Answer

It can be added with another vector.

Question 20

Which of the following is equal to the dot product of vectors A = 3i + 4j and B = 5i - 2j?

Accepted Answer

7

Answer

-7

Answer

11

Answer

19

Question 21

Which of the following operations is NOT applicable to vectors?

Accepted Answer

Multiplication of two vectors together.

Answer

Subtraction of two vectors.

Answer

Dot product of two vectors.

Answer

Addition of two vectors.

Question 22

Which of the following is a unit vector in the direction of vector ?

Accepted Answer

Vector / magnitude of  vector

Answer

Vector - magnitude of  vector

Answer

Vector * magnitude of  vector

Question 23

What is the position vector of a point  relative to the origin?

Accepted Answer

Vector drawn from the origin to the point

Answer

Magnitude of the vector from the origin to the point

Answer

Vector drawn from the point to the origin

Question 24

Which of the following is a scalar quantity?

Accepted Answer

Speed

Answer

Acceleration

Answer

Velocity

Answer

Displacement

Question 25

What is the acceleration due to gravity on Earth?

Accepted Answer

9.8 m/s^2 downward

Answer

9.8 m/s^2 upward

Answer

10 m/s^2 downward

Answer

10 m/s^2 upward

Question 26

Which of the following is NOT a property of vectors?

Accepted Answer

Vectors have a fixed position in space

Answer

Vectors have both magnitude and direction

Answer

Vectors can be added and subtracted

Answer

Vectors can be multiplied by a scalar

Question 27

The dot product of two perpendicular vectors is always:

Accepted Answer

Zero

Answer

The sum of their magnitudes

Answer

The product of their magnitudes

Answer

One

Question 28

Two vectors, A and B, are given as A = (2, 3) and B = (-1, 4). What is the sum of these two vectors, A + B?

Accepted Answer

(1, 7)

Answer

(-2, 1)

Answer

(1, 1)

Answer

(3, 7)

Question 29

A vector has a magnitude of 8 units and makes an angle of 30 degrees with the horizontal axis. What are its horizontal and vertical components?

Accepted Answer

Horizontal component: 4√3, Vertical component: 4

Answer

Horizontal component: 4, Vertical component: 4√3

Answer

Horizontal component: 4√3, Vertical component: 8

Answer

Horizontal component: 8, Vertical component: 4

Question 30

Two vectors, A and B, are given as A = (2, 3) and B = (-1, 4). What is the dot product of these two vectors, A · B?

Accepted Answer

10

Answer

-1

Answer

14

Answer

7

Question 31

A force of 20 N is applied to an object at an angle of 60 degrees with the horizontal. What is the horizontal component of the force?

Accepted Answer

10 N

Answer

20 N

Answer

10√3 N

Answer

17.32 N

Question 32

A projectile is launched horizontally with an initial velocity of 20 m/s. What is the vertical component of its velocity after 2 seconds?

Accepted Answer

19.6 m/s downwards

Answer

0 m/s

Answer

20 m/s downwards

Answer

19.6 m/s upwards

Question 33

What is the angle between two vectors if their dot product is zero?

Accepted Answer

90 degrees

Answer

180 degrees

Answer

0 degrees

Answer

45 degrees

Question 34

Which of the following is NOT a property of vectors?

Accepted Answer

Vectors have a fixed position in space.

Answer

Vectors can be multiplied by scalars.

Answer

Vectors have both magnitude and direction.

Answer

Vectors can be added and subtracted.

Question 35

Two forces of equal magnitude act on an object at an angle of 120 degrees to each other. What is the magnitude of the resultant force?

Accepted Answer

Equal to the magnitude of each individual force

Answer

Twice the magnitude of each individual force

Answer

Zero

Answer

Half the magnitude of each individual force

Question 36

Given vectors a = (2, 3) and b = (4, 5), find the vector a + b.

Accepted Answer

(6, 8)

Answer

(10, 8)

Answer

(8, 10)

Answer

(2, 8)

Question 37

Given vectors a = (2, -1) and b = (3, 4), find the dot product of a and b.

Accepted Answer

11

Answer

-11

Answer

7

Answer

-7

Question 38

A force of 10 Newtons acts on an object in the direction of the vector (3, 4). Find the magnitude of the force.

Accepted Answer

5

Answer

10

Answer

20

Answer

15

Question 39

Given vectors a = (2, 3) and b = (4, -1), find the angle between a and b.

Accepted Answer

90 degrees

Answer

180 degrees

Answer

135 degrees

Answer

45 degrees

Question 40

A particle moves in the direction of the vector (2, 3) with a speed of 5 m/s. Find the particle's displacement after 2 seconds.

Accepted Answer

(10, 15)

Answer

(5, 10)

Answer

(10, 5)

Answer

(15, 10)

Question 41

Given vectors a = (2, 3) and b = (4, 5), find the vector a - b.

Accepted Answer

(-2, -2)

Answer

(-2, 2)

Answer

(2, -2)

Answer

(2, 2)

Question 42

A force of 10 Newtons is applied to an object in the direction of the vector (3, 4). Find the work done by the force if the object moves 5 meters in the direction of the force.

Accepted Answer

50 Joules

Answer

25 Joules

Answer

75 Joules

Answer

100 Joules

Question 43

Given vectors a = (2, 3) and b = (4, 5), find the scalar projection of a onto b.

Accepted Answer

2

Answer

6

Answer

8

Answer

4

Question 44

A particle moves in the direction of the vector (2, 3) with a speed of 5 m/s. Find the particle's acceleration if its velocity changes from (5, 10) to (10, 15) in 2 seconds.

Accepted Answer

(2.5, 2.5)

Answer

(1.25, 1.25)

Answer

(10, 10)

Answer

(5, 5)

Question 45

Given vectors a = (2, 3) and b = (4, 5), find the cross product of a and b.

Accepted Answer

(1, -2)

Answer

(-1, 2)

Answer

(2, 1)

Answer

(-2, -1)

Question 46

Which of the following correctly calculates the magnitude of vector AB, where A = (2, -3) and B = (-1, 4)?

Accepted Answer

√26

Answer

-√26

Answer

√14

Answer

-√14

Question 47

Two forces of 5 N and 10 N act on an object at an angle of 60 degrees to each other. What is the magnitude of the resultant force?

Accepted Answer

13.66 N

Answer

8.66 N

Answer

6.66 N

Answer

18.66 N

Question 48

Which of the following is the dot product of vectors A = (2, 3) and B = (-4, 5)?

Accepted Answer

-2

Answer

-10

Answer

10

Answer

2

Question 49

A vector has an x-component of -5 and a y-component of 12. What is its direction angle measured from the x-axis?

Accepted Answer

-67.4°

Answer

-22.6°

Answer

67.4°

Answer

22.6°

Question 50

Which of the following is the vector equation of a line passing through points A(2, -1) and B(6, 3)?

Accepted Answer

r = (2, -1) + t(4, 4)

Answer

r = (2, -1) + t(2, 2)

Answer

r = (6, 3) + t(4, 4)

Answer

r = (6, 3) + t(2, 2)

Question 51

A car travels 50 km east and then 30 km north. What is the magnitude of its displacement?

Accepted Answer

58.3 km

Answer

80 km

Answer

20 km

Answer

38.3 km

Question 52

The scalar product of two vectors is 0. What does this imply about the vectors?

Accepted Answer

They are perpendicular to each other.

Answer

They have opposite directions.

Answer

They are collinear.

Answer

They have the same magnitude.

Question 53

A force of 100 N is applied to an object at an angle of 30 degrees above the horizontal. What is the vertical component of the force?

Accepted Answer

50 N

Answer

25 N

Answer

75 N

Answer

100 N

Question 54

Which of the following is the unit vector in the direction of vector A = (3, 4)?

Accepted Answer

(3/5, 4/5)

Answer

(5/3, 5/4)

Answer

(4/3, 3/4)

Answer

(3, 4)

Question 55

A projectile is launched with a velocity of 20 m/s at an angle of 45 degrees to the horizontal. What is the horizontal component of its velocity?

Accepted Answer

14.14 m/s

Answer

7.07 m/s

Answer

28.28 m/s

Answer

10.00 m/s