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.

Accepted Answer

The magnitude of a three-dimensional vector is always a non-negative value, calculated as the square root of the sum of the squares of its components (rule C). It cannot be negative, which contradicts choice D. Choices A, B, and C correctly describe properties of a vector's magnitude.

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.

Accepted Answer

The magnitude of the resultant vector from adding two vectors cannot exceed the sum of their magnitudes, adhering to the triangle inequality principle. The resultant's magnitude depends on the vectors' directions and magnitudes, but it cannot be negative, as magnitudes are always non-negative.

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.

Accepted Answer

The dot product is defined as the magnitude of A times the magnitude of B times the cosine of the angle between them, not the sine.

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.

Accepted Answer

The magnitude of the cross product is given by the magnitude of A times the magnitude of B times the sine of the angle between them, not the cosine.

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.

Accepted Answer

The dot product (A · B) of two vectors results in a scalar, not a vector. All other options (A + B, A - B, A × B) result in vectors.

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.

Accepted Answer

The answer of Vector A points horizontally toward the right...

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.

Accepted Answer

The dot product (A · B), the cross product magnitude (|A $\times$ B|), and the magnitudes of vectors (|A| and |B|) are invariant under rotation of the coordinate system. However, the components of vectors A and B themselves will change in a rotated coordinate system, as these depend on the basis vectors of the coordinate system.

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.

Accepted Answer

When vectors are joined head to tail, the angle between their lengths is not the angle between the vectors themselves.

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.

Accepted Answer

The cross product A × B is calculated as the determinant of a matrix with i, j, k unit vectors in the first row, components of A in the second row, and components of B in the third row. This results in 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) $\sqrt { ( 81 + 36 + 16 ) } = \sqrt { 133 }$

Accepted Answer

The answer of If A = 2i + 3j, and...

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.

Accepted Answer

The dot product A · B is calculated as (2i + 3j) · (2j + 3k) = 2i·2j + 2i·3k + 3j·2j + 3j·3k. Since i·j, i·k, and j·k are orthogonal and their dot products are zero, only 3j·2j remains, which equals 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.

Accepted Answer

The cross product $\mathbf{A} \times \mathbf{B} = 0$ implies that vectors $\mathbf{A}$ and $\mathbf{B}$ are parallel, as the cross product is zero only when the vectors are parallel or one of them is zero, but both are non-zero here.

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.

Accepted Answer

The dot product of two non-zero vectors is zero if and only if the vectors are perpendicular to each other.

Question 14

The properties of a vector include
A) magnitude.
B) direction.
C) both of the above.
D) neither of the above.

Accepted Answer

Vectors are defined by both their magnitude (how long they are) and their direction (which way they point).

Question 15

The properties of a scalar include
A) magnitude.
B) direction.
C) both of the above.
D) neither of the above.

Accepted Answer

Scalars are quantities that are fully described by a magnitude (or numerical value) alone and do not include direction, unlike vectors which have both magnitude and direction.

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.

Accepted Answer

The magnitude of a vector $\vec{A}$ with components $A_x$ and $A_y$ is given by $\sqrt{A_x^2 + A_y^2}$. Substituting $A_x = 3.0$ and $A_y = 4.0$, we get $\sqrt{3.0^2 + 4.0^2} = \sqrt{9 + 16} = \sqrt{25} = 5.0$.

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

Accepted Answer

The x-component of a vector in a two-dimensional rectangular coordinate system is given by the magnitude of the vector times the cosine of the angle it makes with the x-axis, which is |A| cos θ.

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

Accepted Answer

A_x is the component of vector A along the x-axis. Since the angle $\theta$ is with respect to the y-axis, A_x is given by |A| sin $\theta$.

Question 19

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

Accepted Answer

The magnitude of vector A, |A|, is related to its x-component, A_x, by the cosine of the angle $\theta$. Therefore, |A| = A_x / 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.

Accepted Answer

Vector A = 2i + 5j is parallel to any vector that is a scalar multiple of it. Vector 4i + 10j can be obtained by multiplying vector A by 2 (2*(2i + 5j) = 4i + 10j), making it parallel to A.

Quiz 3: Vectors

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

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

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

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

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

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

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

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

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

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

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

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

The properties of a vector include

The properties of a scalar include

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

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_xis

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

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

Quiz 3: Vectors

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

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

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

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

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

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

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

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

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