Master Vector Mathematics: Enhance Your Mechanical Engineering Skills

Question 1

Which of the following operations is NOT commutative for vectors?

Accepted Answer

Cross product

Answer

Addition

Answer

Subtraction

Answer

Dot product

Question 2

The dot product of two vectors is a:

Accepted Answer

Scalar quantity

Answer

Vector quantity

Answer

Matrix quantity

Answer

Tensor quantity

Question 3

Which of the following is a unit vector?

Accepted Answer

A vector with a magnitude of 1

Answer

A vector with a magnitude of 0

Answer

A vector with a direction of 0 degrees

Answer

A vector that is parallel to the x-axis

Question 4

The angle between the vectors (1, 0, 0) and (0, 1, 0) is:

Accepted Answer

90 degrees

Answer

0 degrees

Answer

45 degrees

Answer

180 degrees

Question 5

The work done by a force F acting on a particle that moves through a displacement d is given by:

Accepted Answer

W = F.d

Answer

W = F × d

Answer

W = d.F

Answer

W = F - d

Question 6

Which of the following is NOT a commutative operation for vectors?

Accepted Answer

Cross product

Answer

Addition

Answer

Subtraction

Answer

Dot product

Question 7

The vector projection of vector A onto vector B is

Accepted Answer

The component of A in the direction of B

Answer

A unit vector in the direction of B

Answer

The difference between A and B

Answer

The dot product of A and B

Question 8

Which of the following is a unit vector in the direction of vector A = [1, 2, 3]?

Accepted Answer

[1/6, 1/3, 1/2]

Answer

[1/3, 2/3, 3/3]

Answer

[1, 2, 3]

Answer

[6, 3, 2]

Question 9

Represent the position vector of point P with coordinates (2, 3) relative to the origin.

Accepted Answer

2i + 3j

Answer

3i + 2j

Answer

2j + 3i

Answer

3j + 2i

Question 10

Which of the following is NOT a vector operation?

Accepted Answer

Multiplication

Answer

Addition

Answer

Subtraction

Answer

Dot product

Question 11

Which of the following is the cross product of vectors A = i + j and B = i - j?

Accepted Answer

2k

Answer

-2k

Answer

i + j

Answer

-i + j

Question 12

A particle moves along a straight line with a velocity of 5 m/s. What is its displacement after 10 seconds?

Accepted Answer

10 m

Answer

25 m

Answer

50 m

Answer

100 m

Question 13

Which of the following is a scalar quantity?

Accepted Answer

Mass

Answer

Velocity

Answer

Acceleration

Answer

Force

Question 14

Which of the following is the formula for the dot product of two vectors, a and b?

Accepted Answer

a.b = |a|*|b|*cos(theta)

Answer

a.b = |a|*|b|

Answer

a.b = |a|*|b|/cos(theta)

Answer

a.b = |a| - |b|

Question 15

Find the derivative of the vector function **r**(t) = (t^2, e^t, sin(t)).

Accepted Answer

(2t, e^t, cos(t))

Answer

(t^2, e, sin(t))

Answer

((2t)^2, (e^t)^2, (sin(t))^2)

Answer

(t^3, e, cos(t))

Question 16

Which equation accurately represents the dot product of vectors A and B?

Accepted Answer

A · B = |A||B|cosθ

Answer

A · B = |A| + |B|

Answer

A · B = |A| - |B|

Answer

A · B = |A||B|sinθ

Question 17

A force of 100 Newtons is applied to an object with a mass of 10 kilograms. Calculate the acceleration of the object.

Accepted Answer

10 meters per second squared

Answer

1 meter per second squared

Answer

100 meters per second squared

Answer

1000 meters per second squared

Question 18

Determine the unit vector pointing in the opposite direction of V (3, 4, 0).

Accepted Answer

(-3/5, -4/5, 0)

Answer

(3/5, 4/5, 0)

Answer

(3, 4, 0)

Answer

(1, 1, 0)

Question 19

Find the position vector of the midpoint of the line segment connecting points P(1, 2, 3) and Q(5, 6, 7).

Accepted Answer

(3, 4, 5)

Answer

(2, 4, 5)

Answer

(2, 5, 4)

Answer

(5, 4, 5)

Question 20

Identify the scalar quantity:

Accepted Answer

Distance

Answer

Velocity

Answer

Acceleration

Answer

Force

Question 21

In vector algebra, which operation is commutative?

Accepted Answer

Addition

Answer

Subtraction

Answer

Scalar Product (Dot Product)

Answer

Vector Product (Cross Product)

Question 22

The position vector that identifies the location of point P in relation to the origin O is denoted as:

Accepted Answer

OP

Answer

PO

Answer

OP - PO

Answer

PO - OP

Question 23

To determine the vector equation of a straight line passing through points A(x1, y1) and B(x2, y2), use the following equation:

Accepted Answer

r = A + t(B - A)

Answer

r = B + t(A - B)

Answer

r = t(A - B)

Answer

r = t(B - A) + A

Question 24

The angle between vectors A and B can be calculated using:

Accepted Answer

Dot Product

Answer

Cross Product

Answer

Scalar Product

Answer

Vector Product

Question 25

The unit vector that points in the same direction as vector A is:

Accepted Answer

A / |A|

Answer

|A| / A

Answer

A - |A|

Answer

|A| - A

Question 26

Which of the following operations is not applicable to vectors?

Accepted Answer

Exponentiation by a scalar

Answer

Addition

Answer

Subtraction

Answer

Multiplication by a scalar

Question 27

A unit vector is a vector with:

Accepted Answer

Magnitude equal to 1

Answer

Zero magnitude

Answer

Magnitude greater than 1

Answer

Magnitude less than 1

Question 28

The projection of one vector onto another is:

Accepted Answer

The component of the first vector that is parallel to the second vector

Answer

The component of the first vector that is perpendicular to the second vector

Answer

The sum of the two vectors

Answer

The difference of the two vectors

Question 29

In vector analysis, what is a vector defined as?

Accepted Answer

A quantity possessing both magnitude and direction

Answer

A quantity having only magnitude

Answer

A quantity having only direction

Answer

A quantity lacking both magnitude and direction

Question 30

For given vectors A and B, what formula accurately determines the magnitude of their vector sum, A + B?

Accepted Answer

√(||A||² + ||B||² + 2||A||||B||cosθ)

Answer

A + B

Answer

||A|| + ||B||

Answer

√(||A||² + ||B||² - 2||A||||B||cosθ)

Question 31

When two perpendicular vectors are subjected to a dot product operation, what is the resulting value?

Accepted Answer

0

Answer

1

Answer

||A||||B||

Answer

Undefined

Question 32

If two vectors, A and B, are parallel, what is the result of their cross product?

Accepted Answer

0

Answer

A x B

Answer

B x A

Answer

A . B

Question 33

Which of the following accurately describes a unit vector?

Accepted Answer

A vector with a magnitude of 1

Answer

A vector with a direction of 0°

Answer

A vector with a magnitude of 0

Answer

A vector with a direction of 90°

Question 34

For a line that passes through points P(x1, y1) and Q(x2, y2), what is its vector equation?

Accepted Answer

r = P + t(Q - P)

Answer

r = P - t(Q - P)

Answer

r = Q - t(P - Q)

Answer

r = P + t(P - Q)

Question 35

In vector analysis, what formula accurately determines the area of the parallelogram formed by vectors A and B?

Accepted Answer

||A x B||

Answer

||A|| + ||B||

Answer

||A|| * ||B||

Answer

||A - B||

Question 36

Which option represents a unit vector in the direction of A = 2i - 3j?

Accepted Answer

(2/13)i - (3/13)j

Answer

(3/13)i + (2/13)j

Answer

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

Answer

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

Question 37

Regarding orthogonal vectors, which statement is true about their dot product?

Accepted Answer

It is zero

Answer

It is one

Answer

It is negative one

Answer

It is equal to the length of the vectors

Question 38

In the context of vectors, what is the result of the cross product of two parallel vectors?

Accepted Answer

Zero vector

Answer

Unit vector

Answer

Magnitude of the vectors

Answer

Normal to the plane containing the vectors

Question 39

What is the magnitude of the vector **u** = (2, -3, 4)?

Accepted Answer

√29

Answer

√11

Answer

√61

Answer

√12

Question 40

Find the unit vector in the direction of the vector **v** = (1, 2, -3).

Accepted Answer

(1/√14, 2/√14, -3/√14)

Answer

(2/√14, 1/√14, -3/√14)

Answer

(1/√2, 2/√2, -3/√2)

Answer

(3/√14, 1/√14, -2/√14)

Question 41

Which of the following operations is NOT defined for vectors?

Accepted Answer

Subtraction of magnitudes

Answer

Cross product

Answer

Dot product

Answer

Addition

Question 42

Two forces, **F1** = (3i + 4j) N and **F2** = (-2i + 5j) N, act on an object. What is the resultant force?

Accepted Answer

(i + 9j) N

Answer

(5i - j) N

Answer

(5i + 9j) N

Answer

(i + j) N

Question 43

What is the magnitude of the vector **A** = (4i - 3j)?

Accepted Answer

5

Answer

25

Answer

7

Answer

1

Question 44

A force of 10 N acts at an angle of 30 degrees to the horizontal. What are the horizontal and vertical components of the force?

Accepted Answer

Horizontal: 8.66 N, Vertical: 5 N

Answer

Horizontal: 5 N, Vertical: 8.66 N

Answer

Horizontal: 10 N, Vertical: 5 N

Question 45

What is the dot product of the vectors **A** = (2i + 3j) and **B** = (4i - j)?

Accepted Answer

5

Answer

11

Answer

8

Answer

-5

Question 46

The angle between two vectors is 60 degrees. If the magnitudes of the vectors are 5 and 10, what is the magnitude of their cross product?

Accepted Answer

25√3

Answer

25

Answer

50√3

Answer

50

Question 47

A plane flies 200 km east and then 150 km north. What is the resultant displacement of the plane?

Accepted Answer

250 km, 36.9° north of east

Answer

250 km, 53.1° north of east

Answer

350 km, 53.1° north of east

Answer

350 km, 36.9° north of east

Question 48

What is the unit vector in the direction of the vector **A** = (6i + 8j)?

Accepted Answer

(0.6i + 0.8j)

Answer

(6i + 8j)

Answer

(10i + 10j)

Answer

(0.8i + 0.6j)

Question 49

The position vectors of two points are A = (2i + 3j) and B = (5i + 7j). What is the vector AB?

Accepted Answer

(3i + 4j)

Answer

(7i + 10j)

Answer

(10i + 14j)

Answer

(-3i - 4j)

Question 50

A torque of 20 Nm is applied to a wrench. If the force applied is 10 N, what is the length of the wrench?

Accepted Answer

2 meters

Answer

0.5 meters

Answer

4 meters

Answer

1 meter