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Calculus 3 : Vectors and Vector Operations

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Normal Vectors

Which of the following is FALSE concerning a vector normal to a plane (in -dimensional space)?

Possible Answers:

Multiplying it by a scalar gives another normal vector to the plane.

All the other answers are true.

It is parallel to any other normal vector to the plane.

The cross product of any two normal vectors to the plane is <0,0,0> .

It is orthogonal to the plane.

Correct answer:

All the other answers are true.

Explanation:

These are all true facts about normal vectors to a plane. (If the surface is not a plane, then a few of these no longer hold.)

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Example Question #2 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 5\\0 \\7 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} -14\\11 \\10 \end{bmatrix}$ , are orthogonal or not.

Possible Answers:

The two vectors are not orthogonal.

The two vectors are orthogonal.

Correct answer:

The two vectors are orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

$\overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}$

To find the dot product of two vectors given the notation

$\overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}$

Simply multiply terms across rows:

$\overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3$

For our vectors, $\overrightarrow{a}=\begin{bmatrix} 5\\0 \\7 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} -14\\11 \\10 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=-70+0+70=0$

The two vectors are orthogonal.

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Example Question #1 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 7\\7 \\3 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}1 \\-1 \\2 \end{bmatrix}$ , are orthogonal or not.

Possible Answers:

The two vectors are not orthogonal.

The two vectors are orthogonal.

Correct answer:

The two vectors are not orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

$\overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}$

To find the dot product of two vectors given the notation

$\overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}$

Simply multiply terms across rows:

$\overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3$

For our vectors, $\overrightarrow{a}=\begin{bmatrix} 7\\7 \\3 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}1 \\-1 \\2 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=7-7+6=6$

The two vectors are not orthogonal.

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Example Question #2 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 11\\8 \\-10 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 2\\3 \\4 \end{bmatrix}$ , are orthogonal or not.

Possible Answers:

The two vectors are not orthogonal.

The two vectors are orthogonal.

Correct answer:

The two vectors are not orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

$\overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}$

To find the dot product of two vectors given the notation

$\overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}$

Simply multiply terms across rows:

$\overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3$

For our vectors, $\overrightarrow{a}=\begin{bmatrix} 11\\8 \\-10 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 2\\3 \\4 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=22+24-40=6$

The two vectors are not orthogonal.

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Example Question #2501 : Calculus 3

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 13\\5 \\12 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 2\\2 \\-3 \end{bmatrix}$ , are orthogonal or not.

Possible Answers:

The two vectors are orthogonal.

The two vectors are not orthogonal.

Correct answer:

The two vectors are orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

$\overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}$

To find the dot product of two vectors given the notation

$\overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}$

Simply multiply terms across rows:

$\overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3$

For our vectors, $\overrightarrow{a}=\begin{bmatrix} 13\\5 \\12 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 2\\2 \\-3 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=26+10-36=0$

The two vectors are orthogonal.

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Example Question #501 : Vectors And Vector Operations

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 11\\-8 \\2 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 2\\ 1\\ -7\end{bmatrix}$ , are orthogonal or not.

Possible Answers:

The two vectors are not orthogonal.

The two vectors are orthogonal.

Correct answer:

The two vectors are orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

$\overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}$

To find the dot product of two vectors given the notation

$\overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}$

Simply multiply terms across rows:

$\overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3$

For our vectors, $\overrightarrow{a}=\begin{bmatrix} 11\\-8 \\2 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 2\\ 1\\ -7\end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=22-8-14=0$

The two vectors are orthogonal.

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Example Question #12 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 3\\0 \\3 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 0\\0 \\1 \end{bmatrix}$ , are orthogonal or not.

Possible Answers:

The two vectors are not orthogonal.

The two vectors are orthogonal.

Correct answer:

The two vectors are not orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

$\overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}$

To find the dot product of two vectors given the notation

$\overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}$

Simply multiply terms across rows:

$\overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3$

For our vectors, $\overrightarrow{a}=\begin{bmatrix} 3\\0 \\3 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 0\\0 \\1 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=0+0+3=3$

The two vectors are not orthogonal.

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Example Question #2502 : Calculus 3

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 4\\ 9\\ 0\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 0\\ 0\\-6 \end{bmatrix}$ , are orthogonal or not.

Possible Answers:

The two vectors are not orthogonal.

The two vectors are orthogonal.

Correct answer:

The two vectors are orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

$\overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}$

To find the dot product of two vectors given the notation

$\overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}$

Simply multiply terms across rows:

$\overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3$

For our vectors, $\overrightarrow{a}=\begin{bmatrix} 4\\ 9\\ 0\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 0\\ 0\\-6 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=0+0+0=0$

The two vectors are orthogonal.

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Example Question #2511 : Calculus 3

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 14\\1 \\15 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}1 \\1 \\-1 \end{bmatrix}$ , are orthogonal or not.

Possible Answers:

The two vectors are not orthogonal.

The two vectors are orthogonal.

Correct answer:

The two vectors are orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

$\overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}$

To find the dot product of two vectors given the notation

$\overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}$

Simply multiply terms across rows:

$\overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3$

For our vectors, $\overrightarrow{a}=\begin{bmatrix} 14\\1 \\15 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}1 \\1 \\-1 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=14+1-15=0$

The two vectors are orthogonal.

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Example Question #501 : Vectors And Vector Operations

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 6\\ 2\\3 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 1\\ -4\\ 1\end{bmatrix}$ , are orthogonal or not.

Possible Answers:

The two vectors are not orthogonal.

The two vectors are orthogonal.

Correct answer:

The two vectors are not orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

$\overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}$

To find the dot product of two vectors given the notation

$\overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}$

Simply multiply terms across rows:

$\overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3$

For our vectors, $\overrightarrow{a}=\begin{bmatrix} 6\\ 2\\3 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 1\\ -4\\ 1\end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=6-8+3=1$

The two vectors are not orthogonal.

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Calculus 3 : Vectors and Vector Operations

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #1 : Normal Vectors

Example Question #2 : Normal Vectors

Example Question #1 : Normal Vectors

Example Question #2 : Normal Vectors

Example Question #2501 : Calculus 3

Example Question #501 : Vectors And Vector Operations

Example Question #12 : Normal Vectors

Example Question #2502 : Calculus 3

Example Question #2511 : Calculus 3

Example Question #501 : Vectors And Vector Operations

All Calculus 3 Resources

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