Question 1

The angle between the diagonal of a cube and an edge is

Accepted Answer

A)  45.0°. 
B)  54.7°. 
C)  90.0°. 
D)  35.3°. 
A)  45.0°. 
B)  54.7°. 
C)  90.0°. 
D)  35.3°. B

Question 2

The dot product (A · B) of two vectors can be correctly thought of in all of the following ways except

Accepted Answer

A)  the magnitude of A times the component of B in the direction of A. 
B)  the magnitude of B times the component of A in the direction of B. 
C)  the magnitude of A times the magnitude of B times the cos of the angle between A and B. 
D)  the magnitude of A times the magnitude of B times the sin of the angle between A and B. 
A)  the magnitude of A times the component of B in the direction of A. 
B)  the magnitude of B times the component of A in the direction of B. 
C)  the magnitude of A times the magnitude of B times the cos of the angle between A and B. 
D)  the magnitude of A times the magnitude of B times the sin of the angle between A and B. D

Question 3

If A = -2i - 4j + 4k and B = i - 2j + 2k, the vector of magnitude 2 that is perpendicular to both A and B is

Accepted Answer

A)  (j + k). 
B)  $\sqrt { 2 ( \mathbf { j } - \mathbf { k } ) }$ 
C)  $\sqrt { 2 ( \mathbf { j } + \mathbf { k } ) }$ 
D)  (j - k). 
A)  (j + k). 
B)  $\sqrt { 2 ( \mathbf { j } - \mathbf { k } ) }$ 
C)  $\sqrt { 2 ( \mathbf { j } + \mathbf { k } ) }$ 
D)  (j - k). C

Question 4

If A = 2i + 3j, and B = 2j + 3k, then A $\times$ B is equal to

Accepted Answer

A)  2i + 5j + 3k. 
B)  2i - 5j + 3k. 
C)  9i - 6j + 4k. 
D)  9i + 6j + 4k. 
A)  2i + 5j + 3k. 
B)  2i - 5j + 3k. 
C)  9i - 6j + 4k. 
D)  9i + 6j + 4k.

Question 5

If A = 2i - j - 2k and B = i - 2j + 3k, the dot product A · B is given by

Accepted Answer

A)  0. 
B)  2. 
C)  -8. 
D)  none of the above. 
A)  0. 
B)  2. 
C)  -8. 
D)  none of the above.

Question 6

The magnitude of the resultant vector obtained by adding any two vectors

Accepted Answer

A)  is always at least as large as the magnitude of either vector. 
B)  is never larger than the simple arithmetic sum of the magnitudes of the two vectors. 
C)  is always the square root of the sum of the squares of the magnitudes of the two vectors. 
D)  may be positive or negative, depending on the directions of the two vectors. 
A)  is always at least as large as the magnitude of either vector. 
B)  is never larger than the simple arithmetic sum of the magnitudes of the two vectors. 
C)  is always the square root of the sum of the squares of the magnitudes of the two vectors. 
D)  may be positive or negative, depending on the directions of the two vectors.

Question 7

Points (0,0,0), (1,1,0), and (0,1,1) belong to the same plane. The vector that is perpendicular to the plane is

Accepted Answer

A)  j + k. 
B)  i. 
C)  i - j - k. 
D)  i - j + k. 
A)  j + k. 
B)  i. 
C)  i - j - k. 
D)  i - j + k.

Question 8

Consider the vector A = 2i + 5j. Vectors parallel to A include

Accepted Answer

A)  4i + 10j. 
B)  5i + 2j. 
C)  5i - 2j. 
D)  none of the above. 
A)  4i + 10j. 
B)  5i + 2j. 
C)  5i - 2j. 
D)  none of the above.

Question 9

If A = 2i + 3j, and B = 2j + 3k, then A · B is equal to

Accepted Answer

A)  4 + 6 + 6 + 9 = +25. 
B)  4 - 6 + 6 + 9 = +13. 
C)  0 + 0 + 6 + 0 = +6. 
D)  0 + 0 - 6 + 0 = -6. 
A)  4 + 6 + 6 + 9 = +25. 
B)  4 - 6 + 6 + 9 = +13. 
C)  0 + 0 + 6 + 0 = +6. 
D)  0 + 0 - 6 + 0 = -6.

Question 10

If vector A makes an angle of $\theta$with respect to the x axis and Ax is given in a two-dimensional rectangular coordinate system, then |A| is equal to Ax times

Accepted Answer

A)  sin $\theta$. 
B)  l/(sin $\theta$). 
C)  tan$\theta$. 
D)  l/(cos $\theta$). 
A)  sin $\theta$. 
B)  l/(sin $\theta$). 
C)  tan$\theta$. 
D)  l/(cos $\theta$).

Question 11

$$\text { Determine the result for the following operation: } ( \mathbf { k } \times \mathbf { i } ) \times ( \mathbf { j } \times \mathbf { k } ) \text {. }$$

Accepted Answer

A)  -k 
B)  k 
C)  0 
D)  1 
A)  -k 
B)  k 
C)  0 
D)  1

Question 12

If A = 3i - 2j + 2k and B = i + 2j + 2k, the unit vector perpendicular to both vectors A and B is

Accepted Answer

A)  -(2/3)i - (1/3)j + (2/3)k. 
B)  -(2/3)i - (2/3)j - (1/3)k. 
C)  (2/3)i + (1/3)j + (2/3)k. 
D)  none of the above. 
A)  -(2/3)i - (1/3)j + (2/3)k. 
B)  -(2/3)i - (2/3)j - (1/3)k. 
C)  (2/3)i + (1/3)j + (2/3)k. 
D)  none of the above.

Question 13

$$\text { If } \mathbf { A } \cdot \mathbf { B } = 0 \text {, and } | \mathbf { A } | \neq 0 , | \mathbf { B } | \neq 0 \text {, then }$$

Accepted Answer

A)  A is perpendicular to B. 
B)  A is parallel to B. 
C)  either of the first two answers may be correct, depending on details not supplied. 
D)  neither of the first two answers is correct. 
A)  A is perpendicular to B. 
B)  A is parallel to B. 
C)  either of the first two answers may be correct, depending on details not supplied. 
D)  neither of the first two answers is correct.

Question 14

The properties of a scalar include

Accepted Answer

A)  magnitude. 
B)  direction. 
C)  both of the above. 
D)  neither of the above. 
A)  magnitude. 
B)  direction. 
C)  both of the above. 
D)  neither of the above.

Question 15

$$\text { If } \mathbf { A } \times \mathbf { B } = 0 \text {, and } | \mathbf { A } | \neq 0 , | \mathbf { B } | \neq 0 \text {, then }$$

Accepted Answer

A)  A is perpendicular to B. 
B)  A is parallel to B. 
C)  either of the first two answers may be correct, depending on details not supplied. 
D)  neither of the first two answers is correct. 
A)  A is perpendicular to B. 
B)  A is parallel to B. 
C)  either of the first two answers may be correct, depending on details not supplied. 
D)  neither of the first two answers is correct.

Question 16

If A = 2i - 3j + 2k and B = -i + j + 2k, the unit vector in the direction given by the difference A - B is

Accepted Answer

A)  (3/5)i - (4/5)j. 
B)  3i + 4j. 
C)  3i - 4j. 
D)  (3/5)i + (4/5)j. 
A)  (3/5)i - (4/5)j. 
B)  3i + 4j. 
C)  3i - 4j. 
D)  (3/5)i + (4/5)j.

Question 17

If A = 2i - j - 2k and B = i - 2j + 3k, the cross product A $\times$ B is

Accepted Answer

A)  -7i - 8j - 3k. 
B)  -7i + 8j - 3k. 
C)  +7i + 8j + 3k. 
D)  -7i - 8j + 3k. 
A)  -7i - 8j - 3k. 
B)  -7i + 8j - 3k. 
C)  +7i + 8j + 3k. 
D)  -7i - 8j + 3k.

Question 18

A vector A has a magnitude |A| and makes an angle $\theta$ with respect to the x axis in a two-dimensional rectangular coordinate system. Ax is given by

Accepted Answer

A)  |A| sin $\theta$. 
B)  |A| cos $\theta$. 
C)  |A| tan $\theta$. 
D)  none of the above. 
A)  |A| sin $\theta$. 
B)  |A| cos $\theta$. 
C)  |A| tan $\theta$. 
D)  none of the above.

Question 19

In a rectangular coordinate system, the magnitude of a three-dimensional vector obeys all of the following rules except

Accepted Answer

A)  it is at least as large as that of any single component. 
B)  it is smaller than the simple arithmetic sum of the magnitudes of the components. 
C)  it is the square root of the sum of the squares of the individual components. 
D)  it may be positive or negative, depending on the direction of the components. 
A)  it is at least as large as that of any single component. 
B)  it is smaller than the simple arithmetic sum of the magnitudes of the components. 
C)  it is the square root of the sum of the squares of the individual components. 
D)  it may be positive or negative, depending on the direction of the components.

Question 20

The magnitude of the cross product (A $\times$ B) of two vectors can be correctly thought of in all of the following ways except

Accepted Answer

A)  the magnitude of A times the component of B perpendicular to A. 
B)  the magnitude of B times the component of A perpendicular to B. 
C)  the magnitude of A times the magnitude of B times the cos of the angle between A and B. 
D)  the magnitude of A times the magnitude of B times the sin of the angle between A and B. 
A)  the magnitude of A times the component of B perpendicular to A. 
B)  the magnitude of B times the component of A perpendicular to B. 
C)  the magnitude of A times the magnitude of B times the cos of the angle between A and B. 
D)  the magnitude of A times the magnitude of B times the sin of the angle between A and B.

Exam 3: Vectors

The angle between the diagonal of a cube and an edge is

The dot product (A · B) of two vectors can be correctly thought of in all of the following ways except

If A = -2i - 4j + 4k and B = i - 2j + 2k, the vector of magnitude 2 that is perpendicular to both A and B is

If A = 2i + 3j, and B = 2j + 3k, then A ×\times× B is equal to

If A = 2i - j - 2k and B = i - 2j + 3k, the dot product A · B is given by

The magnitude of the resultant vector obtained by adding any two vectors

Points (0,0,0), (1,1,0), and (0,1,1) belong to the same plane. The vector that is perpendicular to the plane is

Consider the vector A = 2i + 5j. Vectors parallel to A include

If A = 2i + 3j, and B = 2j + 3k, then A · B is equal to

If vector A makes an angle of θ\thetaθ with respect to the x axis and Ax is given in a two-dimensional rectangular coordinate system, then |A| is equal to Ax times

Determine the result for the following operation: (k×i)×(j×k). \text { Determine the result for the following operation: } ( \mathbf { k } \times \mathbf { i } ) \times ( \mathbf { j } \times \mathbf { k } ) \text {. } Determine the result for the following operation: (k×i)×(j×k).

If A = 3i - 2j + 2k and B = i + 2j + 2k, the unit vector perpendicular to both vectors A and B is

If A⋅B=0, and ∣A∣≠0,∣B∣≠0, then \text { If } \mathbf { A } \cdot \mathbf { B } = 0 \text {, and } | \mathbf { A } | \neq 0 , | \mathbf { B } | \neq 0 \text {, then } If A⋅B=0, and ∣A∣=0,∣B∣=0, then

The properties of a scalar include

If A×B=0, and ∣A∣≠0,∣B∣≠0, then \text { If } \mathbf { A } \times \mathbf { B } = 0 \text {, and } | \mathbf { A } | \neq 0 , | \mathbf { B } | \neq 0 \text {, then } If A×B=0, and ∣A∣=0,∣B∣=0, then

If A = 2i - 3j + 2k and B = -i + j + 2k, the unit vector in the direction given by the difference A - B is

If A = 2i - j - 2k and B = i - 2j + 3k, the cross product A ×\times× B is

A vector A has a magnitude |A| and makes an angle θ\thetaθ with respect to the x axis in a two-dimensional rectangular coordinate system. Ax is given by

In a rectangular coordinate system, the magnitude of a three-dimensional vector obeys all of the following rules except

The magnitude of the cross product (A ×\times× B) of two vectors can be correctly thought of in all of the following ways except

Space, Time, and Mass

Motion Along a Straight Line

Motion in Two and Three Dimensions

Newtons Laws of Motion

Further Applications of Newtons Laws

Work and Energy

Conservation of Energy

Gravitation

Systems of Particles

Collisions

Rotation of a Rigid Body

Dynamics of a Rigid Body

Statics and Elasticity

Oscillations

Waves

Sound

Fluid Mechanics

The Ideal Gas

Heat

Thermodynamics

Electric Force and the Electric Charge

The Electric Field

Gauss Law

Electrostatic Potential and Energy

Capacitors and Dielectrics

Currents and Ohms Law

Direct Current Circuits

Magnetic Force and Field

Charges and Currents in Magnetic Fields

Electromagnetic Induction

Alternating Current Circuits

Electromagnetic Waves

Reflection, Refraction, and Optics

Interference and Diffraction

The Theory of Special Relativity

Quanta of Light

Spectral Lines, Bohrs Theory, and Quantum Mechanics

Quantum Structure of Atoms, Molecules, and Solids

Nuclei

Elementary Particles and Cosmology

Filters

If A = 2i + 3j, and B = 2j + 3k, then A $\times$ B is equal to

If vector A makes an angle of $\theta$ with respect to the x axis and A_xis given in a two-dimensional rectangular coordinate system, then |A| is equal to A_x times

$\text { Determine the result for the following operation: } ( \mathbf { k } \times \mathbf { i } ) \times ( \mathbf { j } \times \mathbf { k } ) \text {. }$

$\text { If } \mathbf { A } \cdot \mathbf { B } = 0 \text {, and } | \mathbf { A } | \neq 0 , | \mathbf { B } | \neq 0 \text {, then }$

$\text { If } \mathbf { A } \times \mathbf { B } = 0 \text {, and } | \mathbf { A } | \neq 0 , | \mathbf { B } | \neq 0 \text {, then }$

If A = 2i - j - 2k and B = i - 2j + 3k, the cross product A $\times$ B is

A vector A has a magnitude |A| and makes an angle $\theta$ with respect to the x axis in a two-dimensional rectangular coordinate system. A_xis given by

The magnitude of the cross product (A $\times$ B) of two vectors can be correctly thought of in all of the following ways except