Vectors: Definition, Properties, and Applications

Question 1

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

Accepted Answer

-6

Answer

6

Answer

10

Answer

-10

Question 2

Which equation represents the equation of a plane?

Accepted Answer

ax + by + cz = d

Answer

ax + by - cz = d

Answer

ax - by + cz = d

Answer

ax - by - cz = d

Question 3

Determine the vector equation of the line passing through point P(1, 2, 3) and parallel to vector v = (2, 1, -1).

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 4

Which of the following is a necessary condition for two vectors to be orthogonal?

Accepted Answer

Their dot product is zero

Answer

Their cross product is zero

Answer

Their magnitudes are equal

Answer

Their directions are opposite

Question 5

Which of the following is a characteristic of vectors?

Accepted Answer

They have both magnitude and direction.

Answer

They cannot be added or subtracted.

Answer

They are always parallel to each other.

Answer

They have no units.

Question 6

Which of the following operations is used to find the dot product of two vectors?

Accepted Answer

a · b

Answer

a × b

Answer

a + b

Answer

a - b

Question 7

Which of the following is a unit vector in the same direction as the vector v = (2, 3)?

Accepted Answer

(2/√13, 3/√13)

Answer

(-2/√13, -3/√13)

Answer

(3, 2)

Answer

(2, 3)

Question 8

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

Accepted Answer

(3, -5)

Answer

(7, 1)

Answer

(-3, 5)

Answer

(3, 5)

Question 9

Which of the following applications of vectors is used to determine the moment of a force?

Accepted Answer

Cross product

Answer

Vector addition

Answer

Scalar multiplication

Answer

Dot product

Question 10

What is the angle between the vectors a = (2, 1) and b = (-1, 2)?

Accepted Answer

63.4°

Answer

90°

Answer

45°

Answer

0°

Question 11

Which of the following is a scalar quantity?

Accepted Answer

Temperature

Answer

Force

Answer

Velocity

Answer

Acceleration

Question 12

Find the magnitude of the vector $\overrightarrow{u}=4\hat{i}-3\hat{j}$.

Accepted Answer

5

Answer

4

Answer

6

Question 13

Given the vectors $\overrightarrow{p}=2\hat{i}+3\hat{j}$ and $\overrightarrow{q}=3\hat{i}-\hat{j}$, find the scalar multiple of $\overrightarrow{p}$ with a scalar value of 2.

Accepted Answer

$4\hat{i}+6\hat{j}$

Answer

$2\hat{i}+3\hat{j}$

Answer

$-4\hat{i}-6\hat{j}$

Question 14

Find the cross product of the vectors $\overrightarrow{a}=\hat{i}-2\hat{j}$ and $\overrightarrow{b}=3\hat{i}+\hat{j}$.

Accepted Answer

$-5\hat{k}$

Answer

$5\hat{i}$

Answer

$-5\hat{i}$

Answer

$5\hat{k}$

Question 15

Find the projection of the vector $\overrightarrow{a}=3\hat{i}-2\hat{j}$ onto the vector $\overrightarrow{b}=2\hat{i}+\hat{j}$.

Accepted Answer

$\frac{6}{5}\hat{i}+\frac{3}{5}\hat{j}$

Answer

$\frac{3}{5}\hat{i}-\frac{6}{5}\hat{j}$

Answer

$\frac{6}{5}\hat{i}-\frac{3}{5}\hat{j}$

Answer

$\frac{3}{5}\hat{i}+\frac{6}{5}\hat{j}$

Question 16

The magnitude of the vector \overrightarrow{AB} is defined as the:

Accepted Answer

Length of the vector

Answer

Angle between its components

Answer

Dot product of its components

Answer

Sum of its components

Question 17

Which of the following vectors is a unit vector?

Accepted Answer

\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}

Answer

2\hat{i}-\hat{j}

Answer

\hat{i}+\hat{j}

Question 18

Which of the following is a vector quantity?

Accepted Answer

Velocity

Answer

Mass

Answer

Temperature

Answer

Time

Question 19

What is the position vector of point P with coordinates (2, 3)?

Accepted Answer

2\hat{i}+3\hat{j}

Answer

3\hat{i}-2\hat{j}

Answer

\hat{i}+\hat{j}

Answer

-2\hat{i}+3\hat{j}

Question 20

Which of these statements about vectors is FALSE?

Accepted Answer

A vector's magnitude is independent of its direction.

Answer

Vectors have both magnitude and direction.

Answer

Vectors can be added and subtracted.

Answer

Vectors can be multiplied by scalars.

Question 21

What is the unit vector in the same direction as vector a = 2i + 3j?

Accepted Answer

(2/√13)i + (3/√13)j

Answer

2i + 3j

Answer

(1/√13)i + (1/√13)j

Answer

i + j

Question 22

A particle moves from point A (2, 3) to point B (5, -1). What is the displacement vector of the particle?

Accepted Answer

3i - 4j

Answer

3i + 4j

Answer

5i - 1j

Answer

5i + 1j

Question 23

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

Accepted Answer

90°

Answer

135°

Answer

45°

Question 24

Which of these is NOT a vector quantity?

Accepted Answer

Mass

Answer

Displacement

Answer

Force

Answer

Velocity

Question 25

A particle moves from point A(2, 3) to point B(7, 10). What is the displacement vector of the particle?

Accepted Answer

5**i** + 7**j**

Answer

-5**i** - 7**j**

Answer

7**i** + 5**j**

Answer

-7**i** - 5**j**

Question 26

Which of the following is a practical application of vectors in real life?

Accepted Answer

Calculating forces in structural analysis

Answer

Determining chemical composition

Answer

Predicting weather patterns

Answer

Measuring temperature

Question 27

Which of the following is NOT a property of vectors?

Accepted Answer

Vectors have no magnitude

Answer

Vectors can be added and subtracted

Answer

Vectors have both magnitude and direction

Answer

Vectors can be multiplied by scalars

Question 28

A vector  (\mathbf{v} = (a, b, c)) is called a zero vector when:

Accepted Answer

a, b, and c are all equal to 0

Answer

only b and c are equal to 0

Answer

a, b, and c are all different from 0

Answer

only a is equal to 0

Question 29

Which of the following is a vector quantity, meaning it has both magnitude and direction?

Accepted Answer

Velocity

Answer

Mass

Answer

Energy

Answer

Temperature

Question 30

Two vectors are considered equal if they have the same:

Accepted Answer

Magnitude and direction

Answer

Direction only

Answer

Neither magnitude nor direction

Answer

Magnitude only

Question 31

The sum of two vectors, a and b, is represented by:

Accepted Answer

a + b

Answer

a . b

Answer

a x b

Answer

a - b

Question 32

What is the unit vector in the direction of the vector a = 2i + 3j?

Accepted Answer

(2/√13)i + (3/√13)j

Answer

(2/5)i + (3/5)j

Answer

(1/√13)i + (1/√13)j

Answer

(3/√13)i + (2/√13)j

Question 33

The dot product of two vectors, a and b, is given by the formula:

Accepted Answer

a.b = |a||b|cosθ

Answer

a.b = |a| - |b|

Answer

a.b = |a||b|sinθ

Answer

a.b = |a| + |b|

Question 34

The cross product of two vectors, a and b, results in a vector that is:

Accepted Answer

Perpendicular to both a and b

Answer

Parallel to both a and b

Answer

Zero

Answer

In the same direction as a + b

Question 35

The moment of a force about a point is a vector quantity determined by:

Accepted Answer

The force's magnitude, direction, and the point's position

Answer

The force's magnitude only

Answer

The force's direction only

Question 36

Which of the following applications of vectors is used to model the trajectory of a projectile in motion?

Accepted Answer

Projectile motion

Answer

Electrostatics

Answer

Thermodynamics

Answer

Fluid dynamics

Question 37

Given vector **a** = <2, -3> and vector **b** = <-1, 5>, what is the result of **a** + **b**?

Accepted Answer

<1, 2>

Answer

<3, -8>

Answer

<3, 2>

Answer

<-1, 8>

Question 38

Calculate the dot product of vectors **p** = <2, 1> and **q** = <-3, 4>.

Accepted Answer

-2

Answer

6

Answer

<-6, 4>

Answer

10

Question 39

Two vectors, **u** and **v**, are orthogonal. Which of the following statements is always true?

Accepted Answer

**u** ⋅ **v** = 0

Answer

**u** × **v** = 0

Answer

**u** + **v** = 0

Answer

**u** - **v** = 0

Question 40

What is the magnitude of vector **r** = <-2, 5, 1>?

Accepted Answer

√30

Answer

6

Answer

√6

Answer

30

Question 41

Calculate the cross product of vectors **c** = <1, 0, 2> and **d** = <3, -1, 1>.

Accepted Answer

<2, 5, -1>

Answer

<-1, -5, -2>

Answer

<-2, -5, 1>

Answer

<1, 5, 2>

Question 42

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

Accepted Answer

Scalar Multiplicative Identity

Answer

Commutative Property

Answer

Additive Identity Property

Answer

Associative Property

Question 43

Given points A(2, -1) and B(4, 3), what is the vector represented by AB?

Accepted Answer

<2, 4>

Answer

<6, 2>

Answer

<-2, -4>

Answer

<-6, -2>

Question 44

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

Accepted Answer

5

Answer

1

Answer

7

Answer

-1

Question 45

Which of the following is the unit vector in the direction of v = (2, -1)?

Accepted Answer

(2/√5, -1/√5)

Answer

(1/√5, -2/√5)

Answer

(√5/2, -√5/1)

Answer

(2, -1)

Question 46

If a = (1, 2) and b = (3, -1), what is a + b?

Accepted Answer

(4, 1)

Answer

(1, -2)

Answer

(3, 2)

Answer

(2, -3)

Question 47

What is the dot product of a = (2, 1) and b = (-1, 3)?

Accepted Answer

1

Answer

5

Answer

0

Answer

-5

Question 48

Given a = (1, 2, 3) and b = (4, -1, 0), find the cross product a x b.

Accepted Answer

(3, 12, -9)

Answer

(3, -12, -9)

Answer

(3, 12, 9)

Question 49

Two vectors are orthogonal if their dot product is:

Accepted Answer

0

Answer

undefined

Answer

1

Answer

-1

Question 50

What is the projection of a = (4, 1) onto b = (1, 2)?

Accepted Answer

(2/5, 4/5)

Answer

(4/5, 2/5)

Answer

(1/5, 2/5)

Answer

(2, 4)

Question 51

A plane is defined by the equation 2x + 3y - z = 5. What is the normal vector to this plane?

Accepted Answer

(2, 3, -1)

Answer

(5, 2, 3)

Answer

(2, -3, -1)

Answer

(2, 3, 1)

Question 52

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

Accepted Answer

5√3 N

Answer

10N

Answer

10√3 N

Answer

5N

Question 53

Which of the following statements about vectors is NOT true?

Accepted Answer

Vectors can be added and subtracted like scalars.

Answer

Vectors have both magnitude and direction.

Answer

Vectors can be multiplied by scalars.

Answer

The zero vector has a magnitude of 0.