Deck 3: Vectors

Question 1

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

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.

Answer 1

it may be positive or negative, depending on the direction of the components.

Question 2

The magnitude of the resultant vector obtained by adding any 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.

Answer 2

is never larger than the simple arithmetic sum of the magnitudes of the two vectors.

Question 3

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

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.

Answer 3

the magnitude of A times the magnitude of B times the sin of the angle between A and B.

Question 4

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

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.

Question 5

All of the following combinations of vectors A and B are themselves vectors except

A) A + B.
B) A - B.
C) A · B.
D) A $\times$ B.

Question 6

Vector A points horizontally toward the right of the paper. Vector B points perpendicular to the plane of the paper and toward the reader. The direction of A $\times$ B is

A) in the plane of the paper, toward the top.
B) in the plane of the paper, toward the bottom.
C) in the plane of the paper, toward the left.
D) perpendicular to the plane of the paper.

Question 7

Consider vectors A and B (not parallel to each other). In a rotated coordinate system (compared to an original coordinate system), all of the following remain unchanged except

A) A · B.
B) A $\times$ B.
C) |A| and |B|.
D) the components of A and the components of B.

Question 8

The angle between two vectors is equal to the (smallest) angle between their lengths when they are joined in all of the following ways except

A) tail to tail.
B) head to head.
C) head to tail.
D) Hold it! There are no exceptions.

Question 9

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

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

Question 10

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

A) 4 + 5 + 3 = 12.
B) $\sqrt { ( 16 + 25 + 9 ) } = \sqrt { 50 }$
C) (81 + 36 + 16) = 133.
D) $\sqrt { ( 81 + 36 + 16 ) } = \sqrt { 133 }$

Question 11

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

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 12

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

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 13

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

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 vector include

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

Question 15

The properties of a scalar include

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

Question 16

A vector A has the components A_x = 3.0 and A_y = 4.0. The magnitude of A is

A) 1.0.
B) 3.5.
C) 5.0.
D) 7.0.

Question 17

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_x is given by

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

Question 18

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

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

Question 19

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

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

Question 20

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

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

Question 21

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

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

Question 22

Consider two vectors A and B. |A $\times$ B| = |A| |B|. This implies that

A) A and B are perpendicular.
B) A and B are parallel.
C) the angle between A and B is 45°.
D) Hold it! Such a situation is not possible.

Question 23

If A, B, and C are nonzero vectors, the quantity A · B $\times$ C cannot be

A) zero.
B) negative.
C) a scalar.
D) a vector.

Question 24

All of the following combinations of vectors A, B, and C are scalars except

A) A + B - C.
B) (A - B) · C.
C) A · B $\times$ C.
D) A · B.

Question 25

All of the following combinations of vectors A, B, and C are vectors except

A) A + B + C.
B) (A · B)C.
C) A · B $\times$ C.
D) A $\times$ B.

Question 26

The dot product between vector A and vector B is 1, and the cross product between the two vectors is 2k. The angle between the two vectors

A) is 26.6°.
B) is 45.0°.
C) is 63.4°.
D) cannot be determined from the given information.

Question 27

If A = 2i + 2k and B = i + j, the angle between vector A and vector B is

A) 30°.
B) 45°.
C) 60°.
D) 90°.

Question 28

If A = 2i - 2j + k, B = i + 2j + 2k, and C = - i + 2j + k, the area of a parallelepiped whose edges are defined by the vectors A, B, and C is

A) 3.
B) 6.
C) 8.
D) 9.

Question 29

If A = 4i - 2j + 3k and B = 2i + j - 2k, the angle between vector A and vector B is

A) 0°.
B) 45°.
C) -45°.
D) 90°.

Question 30

If A = i + 3j and B = 3i + j, the angle between vector A and vector B is

A) 53°.
B) 30°.
C) 45°.
D) 37°.

Question 31

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

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

Question 32

If A = 2i - 2j - k and B = i + 2j - 2k, the area of a parallelogram whose sides are defined by vectors A and B is

A) 0.
B) 3.
C) 6.
D) 9.

Question 33

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

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

Question 34

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

A) 0
B) -j
C) -i
D) j

Question 35

If A = -i + 2j + k and B = i + j + 2k, the angle between vectors A and B is

A) 53°.
B) 30°.
C) 37°.
D) 60°.

Question 36

$$\text { The resulting unit vector for the operation } \mathbf { k } \times [ \mathbf { j } \times ( \mathbf { j } \times \mathbf { i } ) ] \text { is }$$

A) the null vector.
B) -j.
C) -i.
D) j.

Question 37

A person walks one block north, two blocks west, and then two blocks northwest. The magnitude and direction of the resultant are

A) 4.2 blocks at an angle of 55° west of north.
B) 4.2 blocks at an angle of 35° west of north.
C) 5.0 blocks at an angle of 55° west of north.
D) 5.0 blocks at an angle of 35° west of north.

Question 38

If A = 2i + j - k, B = 2i - 3j + 3k, R = i - j + 4k, and A + B + C = R, then C is

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

Question 39

If A = i + 2j and B = 2i - j, the angle between vectors A and B is

A) 30°.
B) -45°.
C) 60°.
D) 90°.

Question 40

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

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

Question 41

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

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 42

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

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

Question 43

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

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

Question 44

Let vector A = 3.0i + 2.0j. A new rectangular coordinate system is generated by a 45° counterclockwise rotation about the k axis. Vector A in the new coordinate system is given by

A) 3.5i - 0.71j.
B) 2.0i + 3.0j.
C) 3.5i + 0.71j.
D) 2.0i - 3.0j.

Question 45

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

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

Question 46

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

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

Question 47

Let vector A = -2i - j + 2k. The unit vector parallel to vector A is

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

Question 48

A vector A has a magnitude of 3.0 units and makes an angle of 30° with the + x axis in a two-dimensional rectangular coordinate system. A_y is equal to

A) 1.5 units.
B) -1.5 units.
C) 2.6 units.
D) -2.6 units.

Question 49

If A = -2i - 4j + 4k, B = i + 2j - 2k, and C = 2i - 3j + 2k, then A·(B $\times$ C) is

A) 0.
B) 4i + 24j - 28k.
C) -48.
D) -4i - 24j + 28k.

Question 50

The dot product of a cross product of three vectors A·(B $\times$ C) represents

A) the area of the parallelogram defined by B and C.
B) the area of the parallelogram defined by A and B.
C) the volume of the parallelepiped with sides A, B, and C.
D) none of the above.