Vectors and Transformations: Interactive Learning Quizzes for Healthcare Professionals

Question 1

What is the result of the scalar product of two orthogonal vectors?

Accepted Answer

0

Answer

1

Answer

The magnitude of the vectors

Answer

The angle between the vectors

Question 2

What is the determinant of a matrix that represents a reflection?

Accepted Answer

-1

Answer

0

Answer

1

Answer

2

Question 3

What is the inverse of the translation transformation [][]?

Accepted Answer

The translation transformation with the same magnitude but opposite direction

Answer

The translation transformation with the same magnitude and direction

Answer

The rotation transformation with the same angle

Answer

The reflection transformation with respect to the same line

Question 4

Which of the following is the scalar product of vectors a = 2i + 3j and b = 4i - 5j?

Accepted Answer

11

Answer

2

Answer

-11

Answer

-2

Question 5

Which of the following transformations is not a rigid transformation?

Accepted Answer

Dilation

Answer

Translation

Answer

Rotation

Answer

Reflection

Question 6

Which matrix represents a reflection about the y-axis?

Accepted Answer

[1, 0; 0, -1]

Answer

[-1, 0; 0, 1]

Answer

[0, 1; 1, 0]

Answer

[0, -1; -1, 0]

Question 7

Identify the correct transformation matrix for a 90-degree clockwise rotation about the origin.

Accepted Answer

[[cos(90), sin(90); -sin(90), cos(90)]]

Answer

[[cos(90), -sin(90); sin(90), cos(90)]]

Answer

[[sin(90), cos(90); -cos(90), sin(90)]]

Answer

[[sin(90), -cos(90); cos(90), sin(90)]]

Question 8

Calculate the distance between the points (2, 3) and (-1, 5).

Accepted Answer

8

Answer

2

Answer

4

Answer

16

Question 9

What does the scalar triple product of vectors a, b, and c represent geometrically?

Accepted Answer

The volume of a parallelepiped with edges a, b, and c.

Answer

A vector perpendicular to both vectors a and b.

Answer

The area of a parallelogram with sides a and b.

Question 10

Which of the following is NOT a fundamental vector operation?

Accepted Answer

Matrix multiplication

Answer

Addition

Answer

Subtraction

Answer

Scalar multiplication

Question 11

What is the result of the scalar product of two vectors?

Accepted Answer

A single numerical value

Answer

A vector perpendicular to both original vectors

Answer

A vector parallel to both original vectors

Answer

A matrix representing the transformation of one vector into the other

Question 12

What is the determinant of the matrix [2 3; -1 4]?

Accepted Answer

11

Answer

5

Answer

-11

Answer

15

Question 13

Which of the following is a defining property of vectors?

Accepted Answer

They possess both magnitude and direction

Answer

They can be represented using matrices

Answer

They adhere to the distributive property when multiplied

Answer

They are invariably parallel to coordinate axes

Question 14

What is the cross product of the vectors (1, 2, 3) and (4, 5, 6)?

Accepted Answer

(-3, 6, -3)

Answer

(3, -6, 3)

Answer

(-6, 3, -3)

Answer

(6, -3, 3)

Question 15

To find the matrix representing the composite transformation of two given transformation matrices, A and B, what mathematical operation should be performed?

Accepted Answer

Multiply matrix A by matrix B

Answer

Add matrix A to matrix B

Answer

Subtract matrix B from matrix A

Answer

The matrix representing the composite transformation is always the identity matrix

Question 16

What happens when you multiply a vector (overrightarrow{v}) by a transformation matrix (A)?

Accepted Answer

It can change both the vector's position and direction.

Answer

It only changes the vector's position.

Answer

It only changes the vector's direction.

Question 17

Express the equation of a line in 3D space using vector parametric equations.

Accepted Answer

r = a + tb, where 'a' is a position vector and 'b' is a direction vector.

Answer

Three scalar equations: one for each component (x, y, and z).

Answer

Cross product of two vectors coinciding with the line.

Answer

Dot product of the line with a normal vector to the plane containing the line.

Question 18

What does the determinant of a transformation matrix A signify geometrically when it represents a transformation of the plane?

Accepted Answer

The determinant of A equals the area of the image of a unit square under the transformation represented by A.

Answer

The determinant of A equals the angle of rotation of the transformation represented by A.

Answer

The determinant of A equals the magnitude of the translation vector of the transformation represented by A.

Answer

The determinant of A equals the magnitude of the shear of the transformation represented by A.

Question 19

Which of the following transformations **alters** the size or shape of a figure?

Accepted Answer

Dilation

Answer

Translation

Answer

Rotation

Answer

Reflection

Question 20

Which physical quantity is an example of a vector quantity?

Accepted Answer

Velocity

Answer

Time

Answer

Mass

Answer

Temperature

Question 21

Identify which of the following transformations is a translation.

Accepted Answer

(x, y) -> (x + 2, y - 1)

Answer

(x, y) -> (y, x)

Answer

(x, y) -> (x^2, y^2)

Answer

(x, y) -> (2x, y)

Question 22

Determine the determinant of the matrix that represents a rotation of 90 degrees.

Accepted Answer

1

Answer

2

Answer

-1

Answer

0

Question 23

Calculate the dot product of the vectors (1, 2) and (3, 4).

Accepted Answer

11

Answer

5

Answer

17

Answer

12

Question 24

Identify which of the following is a unit vector.

Accepted Answer

(1/2, √3/2)

Answer

(2, 3)

Answer

(1, 0)

Answer

(0, 1)

Question 25

Calculate the angle between the vectors (1, 1) and (2, 1).

Accepted Answer

45 degrees

Answer

60 degrees

Answer

90 degrees

Answer

30 degrees

Question 26

Identify which of the following is a linear transformation.

Accepted Answer

(x, y) -> (2x + 3y, x - y)

Answer

(x, y) -> (x^2, y^2)

Answer

(x, y) -> (|x|, |y|)

Answer

(x, y) -> (sin(x), cos(y))

Question 27

Calculate the dot product of the vectors (1, 2) and (3, 4).

Accepted Answer

11

Answer

5

Answer

15

Answer

20

Question 28

Determine the transformation matrix that translates a point (x, y) by 2 units in the x-direction and 3 units in the y-direction.

Accepted Answer

[[1, 0, 2], [0, 1, 3], [0, 0, 1]]

Answer

[[0, 1, 2], [1, 0, 3], [0, 0, 1]]

Answer

[[0, 1, 3], [1, 0, 2], [0, 0, 1]]

Answer

[[1, 0, 3], [0, 1, 2], [0, 0, 1]]

Question 29

Find the trace of the matrix [[1, 2], [3, 4]]?

Accepted Answer

5

Answer

7

Answer

2

Answer

10

Question 30

Which of the following is NOT a vector operation?

Accepted Answer

Division

Answer

Addition

Answer

Scalar multiplication

Answer

Dot product

Question 31

Calculate the magnitude of the vector v = (2, -3).

Accepted Answer

√13

Answer

-3

Answer

5

Answer

2

Question 32

Identify the transformation represented by the matrix [1 0; 0 -1].

Accepted Answer

Reflection about the x-axis

Answer

Translation

Answer

Shear

Answer

Rotation

Question 33

Which of the following is a true statement about vectors?

Accepted Answer

Vectors have both magnitude and direction

Answer

Vectors can be multiplied by scalars

Answer

Vectors are always parallel to each other

Answer

Vectors can only be added to other vectors

Question 34

Determine the inverse transformation of a translation by (-2, 3).

Accepted Answer

Translation by (2, -3)

Answer

Rotation by 90 degrees

Answer

Translation by (-3, 2)

Answer

Reflection about the y-axis

Question 35

Which of the following transformations preserves the orientation and shape of a figure?

Accepted Answer

Rotation

Answer

Reflection

Answer

Shear

Question 36

Which of the following physical quantities is a vector quantity?

Accepted Answer

Velocity

Answer

Temperature

Answer

Mass

Answer

Time

Question 37

A triangle with vertices A(1, 2), B(3, 4), and C(5, 2) is transformed by the matrix [2 0]
[0 2]. What is the effect of this transformation?

Accepted Answer

Enlargement by a scale factor of 2 about the origin

Answer

Reflection across the x-axis

Answer

Rotation of 180 degrees about the origin

Question 38

Which of the following is NOT a linear transformation?

Accepted Answer

Translation by a non-zero vector

Answer

Rotation about the origin

Answer

Reflection across the x-axis

Question 39

Find the matrix representation of the transformation that translates a point (x, y) by 3 units to the right and 2 units up.

Accepted Answer

[[1, 0, 3], [0, 1, 2], [0, 0, 1]]

Answer

[[1, 0, 2], [0, 1, 3], [0, 0, 1]]

Answer

[[0, 1, 2], [1, 0, 3], [0, 0, 1]]

Answer

[[0, 1, 3], [1, 0, 2], [0, 0, 1]]

Question 40

Calculate the cross product of the vectors A = (1, 2, 3) and B = (4, 5, 6).

Accepted Answer

(-3, 6, -3)

Answer

(3, 6, 3)

Answer

(-3, -6, 3)

Answer

(3, -6, -3)

Question 41

Determine the slope of the line perpendicular to the line y = 2x + 1.

Accepted Answer

-1/2

Answer

1/2

Answer

2

Answer

-2