Vectors in 2D: Interactive Quizzes and Practice Questions

Question 1

What is the formula for finding the magnitude of vector **a** with components (a_x) and (a_y)?

Accepted Answer

||**a**|| = (sqrt(a_x^2 + a_y^2))

Answer

||**a**|| = (a_x^2 - a_y^2)

Answer

||**a**|| = (sqrt(a_x^2 - a_y^2))

Answer

||**a**|| = (a_x + a_y)

Question 2

What is the magnitude of the vector (-4, 3)?

Accepted Answer

7

Answer

1

Answer

5

Answer

9

Question 3

Find the direction of the vector 2i - 5j.

Accepted Answer

126.6°

Answer

63.4°

Answer

233.4°

Answer

306.6°

Question 4

Find the unit vector in the direction of the vector 3i + 4j.

Accepted Answer

3/5i + 4/5j

Answer

4/5i + 3/5j

Answer

3i + 4j

Answer

4i + 3j

Question 5

Which of the following transformations represents a translation of 4 units to the right and 3 units up?

Accepted Answer

T(x, y) → (x + 4, y + 3)

Answer

T(x, y) → (x - 4, y + 3)

Answer

T(x, y) → (x + 4, y - 3)

Answer

T(x, y) → (x - 4, y - 3)

Question 6

Which of the following expressions represents the unit vector in the direction of vector A?

Accepted Answer

A/|A|

Answer

|A|/A

Answer

A+|A|

Answer

1/A

Question 7

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

Accepted Answer

(1, -1)

Answer

(5, 9)

Answer

(-5, -1)

Answer

(-1, 9)

Question 8

Determine the angle between vectors a = (1, 1) and b = (-1, 1).

Accepted Answer

90 degrees

Answer

45 degrees

Answer

135 degrees

Answer

180 degrees

Question 9

Which operation is not defined for vectors?

Accepted Answer

Division by a vector

Answer

Addition

Answer

Subtraction

Answer

Multiplication by a scalar

Question 10

Which of the following operations is not defined for vectors?

Accepted Answer

Division by a scalar

Answer

Addition

Answer

Subtraction

Answer

Multiplication by a scalar

Question 11

Which of the following applications of vectors is used in physics?

Accepted Answer

Representing forces and velocities

Answer

Calculating distances

Answer

Finding angles

Answer

Solving equations

Question 12

What is the position vector of a point P(2, 3) relative to the origin O(0, 0)?

Accepted Answer

(2, 3)

Answer

(0, 3)

Answer

(2, 0)

Answer

(3, 2)

Question 13

Determine the direction of the vector (5, 0).

Accepted Answer

East

Answer

North

Answer

South

Answer

West

Question 14

Calculate the result of multiplying the vector (2, 3) by a scalar of -2.

Accepted Answer

(-4, -6)

Answer

(4, 6)

Answer

(-2, -3)

Answer

(2, -3)

Question 15

What is the result of multiplying the vector (4, -2) by the scalar 3?

Accepted Answer

(12, -6)

Answer

(6, -3)

Answer

(2, 1)

Answer

(-8, 6)

Question 16

Calculate the angle between the vectors (1, 0) and (0, 1).

Accepted Answer

90°

Answer

0°

Answer

45°

Answer

180°

Question 17

Determine the vector equation of the line passing through the points (2, 3) and (5, 7).

Accepted Answer

r = (2, 3) + t(3, 4)

Answer

r = (5, 7) + t(3, 4)

Answer

r = (2, 3) + t(5, 7)

Answer

r = (5, 7) + t(2, 3)

Question 18

Find the area of the triangle with vertices (0, 0), (3, 4), and (6, 0).

Accepted Answer

12

Answer

6

Answer

18

Answer

24

Question 19

Which of these actions is NOT a vector operation?

Accepted Answer

Division

Answer

Addition

Answer

Subtraction

Answer

Scalar multiplication

Question 20

Which of the following represents the vector sum of vectors a and b?

Accepted Answer

a + b

Answer

a - b

Answer

b - a

Answer

a x b

Question 21

Multiplying a vector by a positive scalar k changes:

Accepted Answer

Magnitude of the vector

Answer

Direction of the vector

Answer

Both direction and magnitude

Answer

Neither direction nor magnitude

Question 22

The vector product of two perpendicular vectors is:

Accepted Answer

0

Answer

1

Answer

Magnitude of the vectors

Answer

Direction of the vectors

Question 23

Which of the following operations results in a scalar value?

Accepted Answer

Scalar multiplication

Answer

Vector addition

Answer

Vector subtraction

Answer

Vector product

Question 24

In a two-dimensional Cartesian coordinate system, the unit vector in the direction of the positive y-axis is:

Accepted Answer

(0, 1)

Answer

(1, 0)

Answer

(1, 1)

Answer

(0, -1)

Question 25

How can the dot product of two non-zero vectors **a** and **b** be used to find the angle θ between them?

Accepted Answer

The dot product of **a** and **b** is **|a| * |b| * cos(θ)**.

Answer

The dot product of **a** and **b** is the sine of the angle between them.

Answer

The dot product of **a** and **b** is the tangent of the angle between them.

Answer

The dot product of **a** and **b** is not related to the angle between them.

Question 26

Determine the result of adding the vectors u = (2, 3) and v = (-1, 5).

Accepted Answer

(1, 8)

Answer

(3, 2)

Answer

(2, 8)

Answer

(1, 2)

Question 27

Identify the unit vector in the direction of the vector v = (3, -4).

Accepted Answer

(3/5, -4/5)

Answer

(4/5, 3/5)

Answer

(3, -4)

Answer

(-4, 3)

Question 28

Find the vector product of the vectors u = (2, 1) and v = (3, -2).

Accepted Answer

-8

Answer

8

Answer

7

Answer

-7

Question 29

Which of the following equations represents a vector that is parallel to the x-axis?

Accepted Answer

y = 0

Answer

x = 0

Answer

y = x

Answer

x = y

Question 30

State the true statement about the zero vector.

Accepted Answer

It has a magnitude of 0 and a direction of 0 degrees.

Answer

It has a magnitude of 1 and a direction of 0 degrees.

Answer

It has a magnitude of 0 and is undefined.

Answer

It has a magnitude of 1 and is undefined.

Question 31

Determine the angle between the vectors (1, 0) and (0, 1).

Accepted Answer

90°

Answer

30°

Answer

45°

Answer

60°

Question 32

A 10 N force acts in the direction of (3, 4). Find its horizontal and vertical components.

Accepted Answer

(6 N, 8 N)

Answer

(8 N, 6 N)

Answer

(4 N, 6 N)

Answer

(6 N, 4 N)

Question 33

Find the position vector of a point located 5 units to the right and 3 units up from the origin.

Accepted Answer

(5, 3)

Answer

(3, 5)

Answer

(-5, 3)

Answer

(-3, 5)

Question 34

Which of the following is a unit vector?

Accepted Answer

A vector with a magnitude of 1

Answer

A vector that is parallel to another vector

Question 35

What is the scalar product of two perpendicular vectors?

Accepted Answer

0

Answer

1

Question 36

Given two vectors, ($\overrightarrow{a}$) and ($\overrightarrow{b}$), what is the geometric interpretation of ($\overrightarrow{a} \times \overrightarrow{b}$)?

Accepted Answer

The area of the parallelogram formed by ($\overrightarrow{a}$) and ($\overrightarrow{b}$)

Answer

The volume of the parallelepiped formed by (\overrightarrow{a}), (\overrightarrow{b}), and the zero vector

Question 37

Which of the following is NOT a property of vectors?

Accepted Answer

Vectors have a fixed length

Answer

Vectors can be added and subtracted

Question 38

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

Accepted Answer

(3/5, 4/5)

Answer

(4/5, 3/5)

Answer

(3, 4)

Answer

(4, 3)

Question 39

A particle moves from point A(2, 3) to point B(8, 6). The displacement vector of the particle is:

Accepted Answer

(6, 3)

Answer

(3, 6)

Answer

(6, 9)

Answer

(9, 6)

Question 40

Which of the following operations is NOT applicable to vectors?

Accepted Answer

Cross product

Answer

Addition

Answer

Subtraction

Answer

Multiplication by a scalar

Question 41

Given vectors a = 2i + 3j and b = 4i - 5j, calculate the magnitude of the vector a + b.

Accepted Answer

√38

Answer

1

Answer

√22

Answer

√50

Question 42

Write the equation of a vector with a magnitude of 5 and a direction of 45°. Express the vector using trigonometric ratios.

Accepted Answer

5(cos 45° i + sin 45° j)

Answer

5(sin 45° i + cos 45° j)

Answer

5(cos 60° i + sin 60° j)

Answer

5(sin 60° i + cos 60° j)

Question 43

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

Accepted Answer

8.66 N

Answer

5 N

Answer

10 N

Answer

17.32 N

Question 44

Which of the following is a property of scalar multiplication of vectors?

Accepted Answer

Changes the magnitude of the vector

Answer

Changes the direction of the vector

Answer

Reverses the direction of the vector

Answer

Does not affect the vector

Question 45

Given vector a = (2, -3), calculate the vector -2a.

Accepted Answer

(-4, 6)

Answer

(4, -6)

Answer

(2, 3)

Answer

(-2, -3)

Question 46

Which of the following operations is NOT valid for vectors?

Accepted Answer

Division by a vector

Answer

Addition

Answer

Subtraction

Answer

Multiplication by a scalar

Question 47

Calculate the magnitude of the vector v = 3i + 4j.

Accepted Answer

7

Answer

5

Answer

9

Answer

11

Question 48

If the vector c has a magnitude of 5 and lies along the positive y-axis, what are its components?

Accepted Answer

5j

Answer

5i

Answer

0i + 5j

Answer

5i + 5j

Question 49

If vectors a and b are perpendicular, what is the value of their dot product?

Accepted Answer

0

Answer

1

Answer

-1

Answer

Cannot be determined

Question 50

Find the vector equation of a straight line passing through the points (1, 2) and (4, 6).

Accepted Answer

r = (1, 2) + t(3, 4)

Answer

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

Answer

r = (1, 2) + t(-3, -4)

Answer

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

Question 51

Calculate the magnitude of the vector -3i + 4j.

Accepted Answer

7

Answer

1

Answer

5

Answer

9

Question 52

If a vector has a magnitude of 5 and direction of $	heta = 30^\circ$, what are its component forms?

Accepted Answer

$(\frac{5\sqrt{3}}{2}, \frac{5}{2})$

Answer

$(5, 0)$

Answer

$(4.33, 2.5)$

Answer

$(5\sqrt{3}, 5\sqrt{3})$

Question 53

A force of 10 N acts at an angle of $30^\circ$ above the horizontal. Find the vertical component of the force.

Accepted Answer

8.66 N

Answer

5 N

Answer

10 N

Answer

17.32 N

Question 54

If $overrightarrow{u} = (a, b)$ and $overrightarrow{v} = (c, d)$, what is the condition for $overrightarrow{u}$ and $overrightarrow{v}$ to be parallel?

Accepted Answer

$\frac{a}{c} = \frac{b}{d}$

Answer

$a = c, b = d$

Answer

$a = -c, b = -d$

Answer

$overrightarrow{u} \cdot overrightarrow{v} = 0$

Question 55

Which of the following formulas gives the area of a parallelogram spanned by vectors $overrightarrow{u}$ and $overrightarrow{v}$?

Accepted Answer

$\frac{1}{2} |overrightarrow{u} 	imes overrightarrow{v}|$

Answer

$|overrightarrow{u}| \cdot |overrightarrow{v}|$

Answer

$|overrightarrow{u} + overrightarrow{v}|$

Answer

$overrightarrow{u} \cdot overrightarrow{v}$

Question 56

Given vectors A = (2, 3) and B = (4, 1), find the magnitude of A + B.

Accepted Answer

2√13

Answer

5

Answer

10

Answer

√26

Question 57

Which of the following operations is not commutative?

Accepted Answer

Multiplication of two vectors

Answer

Vector addition

Answer

Vector subtraction

Answer

Scalar multiplication

Question 58

Which of the following is considered a unit vector?

Accepted Answer

(0, 1)

Answer

(2, 0)

Answer

(3, 4)

Answer

(5, 12)

Question 59

Calculate the vector sum a + b if a = (2, 3) and b = (-1, 4).

Accepted Answer

(1, 7)

Answer

(3, 1)

Answer

(-1, 1)

Answer

(-3, 7)

Question 60

Determine the unit vector in the direction of vector v = (3, 4).

Accepted Answer

(3/5, 4/5)

Answer

(4/5, 3/5)

Answer

(3/2, 4/2)

Answer

(4/2, 3/2)

Question 61

To calculate the magnitude of a vector (\vec{a}), which formula should you use?

Accepted Answer

$\Vert \vec{a} \Vert = \sqrt{a_x^2 + a_y^2}$

Answer

$\Vert \vec{a} \Vert = \sqrt{\vec{a} . \vec{a}}$

Answer

$\Vert \vec{a} \Vert = \frac{1}{\sqrt{a_x^2 + a_y^2}}$

Answer

$\Vert \vec{a} \Vert = a_x + a_y$

Question 62

What happens when a vector is multiplied by a scalar of -2 ?

Accepted Answer

The vector is shrunk by a factor of 2.

Answer

The vector is reflected across the origin.

Answer

The vector is stretched by a factor of 2.

Answer

The vector remains unchanged.

Question 63

If (\vec{a} = 2\hat{i} + 3\hat{j}), what is the angle (	heta) between (\vec{a}) and the positive (x)-axis?

Accepted Answer

45 degrees

Answer

30 degrees

Answer

60 degrees

Answer

75 degrees

Question 64

Which of the following is NOT an application of vectors in daily life?

Accepted Answer

Predicting the weather

Answer

Describing the motion of a car

Answer

Calculating the force on a baseball

Answer

Analyzing the vibrations of a guitar string