Vector Algebra and Calculus Practice Quiz (11 questions)

Question 1

1. Which set of attributes defines a vector represented by a directed segment?

Origin, direction, and supporting line

Direction, supporting line, and point of application

Sense, magnitude, and point of application

Origin, direction, sense, and magnitude

Explanation

A vector is defined by its origin, direction, sense, and magnitude. A supporting line is not a defining attribute in this list, and point of application does not replace sense or magnitude.

Answer

Origin, direction, sense, and magnitude

Question 2

2. A vector is known to have a direction, sense, and magnitude, but neither its supporting line nor its point of application is specified. How is it classified?

A directed segment with a specified origin

A free vector

A bound vector

A sliding vector

Explanation

A free vector has a specified direction, sense, and magnitude, while its supporting line and point of application are unspecified. A sliding vector has an unfixed point of application, and a bound vector has its elements determined.

Answer

A free vector

Question 3

3. A coordinate basis has pairwise perpendicular basis vectors, but one of those vectors has magnitude 2. Which description applies?

Neither orthogonal nor orthonormal

Direct, but not orthogonal

Orthogonal, but not orthonormal

Orthonormal, but not orthogonal

Explanation

An orthonormal basis requires both mutual perpendicularity and unit magnitudes; this basis meets the perpendicularity condition but has a vector of magnitude 2. The misconception behind “orthonormal” is that perpendicularity alone is sufficient.

Answer

Orthogonal, but not orthonormal

Question 4

4. In a three-dimensional basis, a vector has components $(-2, 4, 1)$ . Which expression represents the vector?

2\vec{e}_1+4\vec{e}_2+\vec{e}_3

-2\vec{e}_1+4\vec{e}_2+\vec{e}_3

-2\vec{e}_2+4\vec{e}_1+\vec{e}_3

-2\vec{e}_1+4\vec{e}_2-\vec{e}_3

Explanation

A vector with components $(x,y,z)$ is expressed as $x\vec{e}_1+y\vec{e}_2+z\vec{e}_3$ , so the components give the coefficients of their corresponding basis vectors. Omitting the negative sign changes the first component, while swapping basis vectors changes which component belongs to each direction.

Answer

$$-2\vec{e}_1+4\vec{e}_2+\vec{e}_3$$

Question 5

5. What is the component-wise rule for adding two vectors in $\mathbb{R}^3$ ?

Multiply corresponding components to form a new vector.

Add the corresponding components to form a new vector.

Add the magnitudes while keeping the first vector's components.

Multiply each component of the first vector by the second vector's magnitude.

Explanation

Vector addition combines corresponding components, so the result is also a vector in $\mathbb{R}^3$ . Multiplying corresponding components is not the stated rule for a vector sum.

Answer

Add the corresponding components to form a new vector.

Question 6

6. When is a system of vectors linearly independent?

When the system contains a zero vector and all other vectors are nonzero

When a linear combination equal to the zero vector forces every coefficient to be zero

When each vector can be written as a linear combination of the remaining vectors

When a linear combination equal to the zero vector has at least one nonzero coefficient

Explanation

Linear independence means the zero-vector relation has only the trivial solution, with every coefficient equal to zero. A nonzero-coefficient relation instead establishes linear dependence.

Answer

When a linear combination equal to the zero vector forces every coefficient to be zero

Question 7

7. What role does the dot product play in relating two vectors geometrically?

It gives a scalar equal to the product of their magnitudes and the cosine of the angle between them.

It gives a unit vector pointing along the direction of their sum.

It gives the distance between their endpoints, independent of their directions.

It gives a vector perpendicular to both, with magnitude based on the sine of their angle.

Explanation

The dot product is a scalar that combines the vectors’ magnitudes with the cosine of the angle between them. The perpendicular vector described by a sine-based magnitude is the role of the vector product, not the dot product.

Answer

It gives a scalar equal to the product of their magnitudes and the cosine of the angle between them.

Question 8

8. How does an orthonormal set of vectors differ from an orthogonal set?

An orthonormal set has unit-length vectors that may be nonperpendicular, whereas an orthogonal set requires pairwise perpendicularity.

An orthonormal set has pairwise perpendicular vectors of unit length, whereas an orthogonal set requires pairwise perpendicularity.

An orthonormal set has pairwise perpendicular vectors with positive dot products, whereas an orthogonal set requires zero magnitudes.

An orthonormal set has pairwise perpendicular vectors of equal length, whereas an orthogonal set requires unit lengths.

Explanation

Orthonormality combines perpendicularity with the requirement that each vector has magnitude one; orthogonality concerns perpendicularity. Equal lengths do not make an orthogonal set orthonormal unless those lengths are one.

Answer

An orthonormal set has pairwise perpendicular vectors of unit length, whereas an orthogonal set requires pairwise perpendicularity.

Question 9

9. What is the consequence of reversing the order of the two vectors in a vector product?

The resulting vector keeps its direction while its magnitude changes sign.

The resulting vector points in the opposite direction while keeping the same magnitude.

The resulting vector becomes parallel to both input vectors.

The resulting vector remains unchanged in both direction and magnitude.

Explanation

The vector product is antisymmetric, so swapping its inputs multiplies the result by $-1$ and reverses its direction. Its magnitude is unchanged because it depends on the magnitudes of the inputs and the sine of the angle between them.

Answer

The resulting vector points in the opposite direction while keeping the same magnitude.

Question 10

10. To evaluate $\vec{A}\wedge(\vec{B}\wedge\vec{C})$ without first computing the inner vector product, which identity should you use?

\vec{A}(\vec{B}\cdot\vec{C})-\vec{C}(\vec{A}\cdot\vec{B})

\vec{B}(\vec{A}\cdot\vec{B})-\vec{C}(\vec{A}\cdot\vec{C})

\vec{C}(\vec{A}\cdot\vec{B})-\vec{B}(\vec{A}\cdot\vec{C})

\vec{B}(\vec{A}\cdot\vec{C})-\vec{C}(\vec{A}\cdot\vec{B})

Explanation

The double vector product can be rewritten as a linear combination of the two inner-product vectors, with coefficients given by the indicated dot products. The expression involving $\vec{A}(\vec{B}\cdot\vec{C})$ is a different arrangement and does not give this identity.

Answer

$$\vec{B}(\vec{A}\cdot\vec{C})-\vec{C}(\vec{A}\cdot\vec{B})$$

Question 11

11. Which expression gives the orthogonal projection of a vector $\vec{V}$ onto a plane whose unit normal is $\vec{n}$ ?

\vec{V}-(\vec{V}\wedge\vec{n})

(\vec{V}\cdot\vec{n})\vec{n}

\vec{V}-(\vec{V}\cdot\vec{n})\vec{n}

\vec{n}-(\vec{V}\cdot\vec{n})\vec{V}

Explanation

The projection into the plane removes the component of $\vec{V}$ along its unit normal, giving $\vec{V}-(\vec{V}\cdot\vec{n})\vec{n}$ . The expression $(\vec{V}\cdot\vec{n})\vec{n}$ is instead the projection onto the normal axis.

Answer

$$\vec{V}-(\vec{V}\cdot\vec{n})\vec{n}$$

Vector Algebra and Calculus Practice Quiz (11 questions)

All questions and answers

1. Which set of attributes defines a vector represented by a directed segment?

2. A vector is known to have a direction, sense, and magnitude, but neither its supporting line nor its point of application is specified. How is it classified?

3. A coordinate basis has pairwise perpendicular basis vectors, but one of those vectors has magnitude 2. Which description applies?

4. In a three-dimensional basis, a vector has components $(-2, 4, 1)$ . Which expression represents the vector?

5. What is the component-wise rule for adding two vectors in $\mathbb{R}^3$ ?

6. When is a system of vectors linearly independent?

7. What role does the dot product play in relating two vectors geometrically?

8. How does an orthonormal set of vectors differ from an orthogonal set?

9. What is the consequence of reversing the order of the two vectors in a vector product?

10. To evaluate $\vec{A}\wedge(\vec{B}\wedge\vec{C})$ without first computing the inner vector product, which identity should you use?

11. Which expression gives the orthogonal projection of a vector $\vec{V}$ onto a plane whose unit normal is $\vec{n}$ ?

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Vector Algebra and Calculus Practice Quiz (11 questions)

All questions and answers

1. Which set of attributes defines a vector represented by a directed segment?

2. A vector is known to have a direction, sense, and magnitude, but neither its supporting line nor its point of application is specified. How is it classified?

3. A coordinate basis has pairwise perpendicular basis vectors, but one of those vectors has magnitude 2. Which description applies?

4. In a three-dimensional basis, a vector has components (−2,4,1)(-2, 4, 1)(−2,4,1). Which expression represents the vector?

5. What is the component-wise rule for adding two vectors in R3\mathbb{R}^3R3?

6. When is a system of vectors linearly independent?

7. What role does the dot product play in relating two vectors geometrically?

8. How does an orthonormal set of vectors differ from an orthogonal set?

9. What is the consequence of reversing the order of the two vectors in a vector product?

10. To evaluate A⃗∧(B⃗∧C⃗)\vec{A}\wedge(\vec{B}\wedge\vec{C})A∧(B∧C) without first computing the inner vector product, which identity should you use?

11. Which expression gives the orthogonal projection of a vector V⃗\vec{V}V onto a plane whose unit normal is n⃗\vec{n}n?

Study with flashcards

Read the study guide

Related classes

Self-Regulation and Learning

Cooperative and Digital Learning

What Psychology Is Like

Electromagnetism and Charge Conservation

Swedish Pronouns: Man and Indefinites

Swedish Pronouns and Possessives

Make your own practice quizzes

4. In a three-dimensional basis, a vector has components $(-2, 4, 1)$ . Which expression represents the vector?

5. What is the component-wise rule for adding two vectors in $\mathbb{R}^3$ ?

10. To evaluate $\vec{A}\wedge(\vec{B}\wedge\vec{C})$ without first computing the inner vector product, which identity should you use?

11. Which expression gives the orthogonal projection of a vector $\vec{V}$ onto a plane whose unit normal is $\vec{n}$ ?