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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:

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

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

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

It is orthogonal to the plane.

All the other answers are true.

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 #5 : 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 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, $\overrightarrow{a}=\begin{bmatrix} 5\\0 \\7 \end{bmatrix}$ 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 #6 : 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 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, $\overrightarrow{a}=\begin{bmatrix} 13\\5 \\12 \end{bmatrix}$ 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 #2501 : Calculus 3

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 orthogonal.

The two vectors are not 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 #11 : Normal Vectors

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 #12 : Normal Vectors

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 #5 : Normal Vectors

Example Question #1 : Normal Vectors

Example Question #6 : Normal Vectors

Example Question #2501 : Calculus 3

Example Question #501 : Vectors And Vector Operations

Example Question #2501 : Calculus 3

Example Question #2502 : Calculus 3

Example Question #11 : Normal Vectors

Example Question #12 : Normal Vectors

All Calculus 3 Resources

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