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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 #41 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 16\\ 3\\1 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 2\\ -10\\ -2\end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} 16\\ 3\\1 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 2\\ -10\\ -2\end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=32-30-2=0$

The two vectors are orthogonal.

Report an Error

Example Question #42 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 5\\1 \\7 \end{bmatrix}$ 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 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, $\overrightarrow{a}=\begin{bmatrix} 5\\1 \\7 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} -1\\ -1\\1 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=-5-1+7=1$

The two vectors are not orthogonal.

Report an Error

Example Question #43 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 2\\11 \\25 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 5\\-1 \\0 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} 2\\11 \\25 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 5\\-1 \\0 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=10-11+0=-1$

The two vectors are not orthogonal.

Report an Error

Example Question #44 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 44\\ 10\\ 2\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 1\\ -4\\2 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} 44\\ 10\\ 2\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 1\\ -4\\2 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=44-40+4=8$

The two vectors are not orthogonal.

Report an Error

Example Question #51 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 45\\2 \\11 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}1 \\-2 \\-4 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} 45\\2 \\11 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}1 \\-2 \\-4 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=45-4-44=-3$

The two vectors are not orthogonal.

Report an Error

Example Question #52 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 3\\4 \\5 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}3 \\4 \\-5 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} 3\\4 \\5 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}3 \\4 \\-5 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=9+16-25=0$

The two vectors are orthogonal.

Report an Error

Example Question #53 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} -1\\ -4\\8 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 1\\2 \\1 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} -1\\ -4\\8 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 1\\2 \\1 \end{bmatrix}$

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

The two vectors are not orthogonal.

Report an Error

Example Question #54 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 9\\40 \\41 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 9\\40 \\ -41\end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} 9\\40 \\41 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 9\\40 \\ -41\end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=81+1600-1681=0$

The two vectors are orthogonal.

Report an Error

Example Question #55 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix}9 \\16 \\25 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 9\\ 16\\-25 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix}9 \\16 \\25 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 9\\ 16\\-25 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=81+256-625=-288$

The two vectors are not orthogonal.

Report an Error

Example Question #56 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 9\\-5 \\4 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 2\\ 2\\ -2\end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} 9\\-5 \\4 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 2\\ 2\\ -2\end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=18-10-8=0$

The two vectors are orthogonal.

Report an Error

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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 #41 : Normal Vectors

Example Question #42 : Normal Vectors

Example Question #43 : Normal Vectors

Example Question #44 : Normal Vectors

Example Question #51 : Normal Vectors

Example Question #52 : Normal Vectors

Example Question #53 : Normal Vectors

Example Question #54 : Normal Vectors

Example Question #55 : Normal Vectors

Example Question #56 : Normal Vectors

All Calculus 3 Resources

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