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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 #57 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 8\\ -7\\ 3\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 4\\1 \\2 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} 8\\ -7\\ 3\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 4\\1 \\2 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=32-7+6=31$

The two vectors are not orthogonal.

Report an Error

Example Question #58 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix}3 \\2 \\-9 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 4\\3 \\2 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix}3 \\2 \\-9 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 4\\3 \\2 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=12+6-18=0$

The two vectors are orthogonal.

Report an Error

Example Question #59 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 1\\ 9\\-9 \end{bmatrix}$ 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, $\overrightarrow{a}=\begin{bmatrix} 1\\ 9\\-9 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 1\\1 \\1 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=1+9-9=1$

The two vectors are not orthogonal.

Report an Error

Example Question #60 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 2\\-6 \\-7 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}3 \\-6 \\6 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} 2\\-6 \\-7 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}3 \\-6 \\6 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=6+36-42=0$

The two vectors are orthogonal.

Report an Error

Example Question #61 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} -4\\-6 \\-7 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 1\\-5 \\4 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} -4\\-6 \\-7 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 1\\-5 \\4 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=-4+30-28=-2$

The two vectors are not orthogonal.

Report an Error

Example Question #551 : Vectors And Vector Operations

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 6\\ -6\\5 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 13\\ -2\\ -18\end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} 6\\ -6\\5 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 13\\ -2\\ -18\end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=78+12-90=0$

The two vectors are orthogonal.

Report an Error

Example Question #552 : Vectors And Vector Operations

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} -1\\5 \\8 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 3\\4 \\ -3\end{bmatrix}$ , are orthogonal or not.

Possible Answers:

The two vectors are not orthogonal.

The two vectors are orthogonal.

Correct answer:

The two vectors are 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, $\overrightarrow{a}=\begin{bmatrix} -1\\5 \\8 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 3\\4 \\ -3\end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=-3+20-24=-7$

The two vectors are not orthogonal.

Report an Error

Example Question #553 : Vectors And Vector Operations

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix}1 \\ -9\\1 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 13\\1 \\-4 \end{bmatrix}$ , are orthogonal or not.

Possible Answers:

The two vectors are orthogonal.

The two vectors are not orthogonal.

Correct answer:

The two vectors are 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, $\overrightarrow{a}=\begin{bmatrix}1 \\ -9\\1 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 13\\1 \\-4 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=13-9-4=0$

The two vectors are orthogonal.

Report an Error

Example Question #554 : Vectors And Vector Operations

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 5\\ 7\\-1 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 2\\1 \\3 \end{bmatrix}$ , are orthogonal or not.

Possible Answers:

The two vectors are orthogonal.

The two vectors are not orthogonal.

Correct answer:

The two vectors are 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, $\overrightarrow{a}=\begin{bmatrix} 5\\ 7\\-1 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 2\\1 \\3 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=10+7-3=14$

The two vectors are not orthogonal.

Report an Error

Example Question #555 : Vectors And Vector Operations

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 3\\-4 \\-4 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 12\\ 3\\6 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} 3\\-4 \\-4 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 12\\ 3\\6 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=36-12-24=0$

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 #57 : Normal Vectors

Example Question #58 : Normal Vectors

Example Question #59 : Normal Vectors

Example Question #60 : Normal Vectors

Example Question #61 : Normal Vectors

Example Question #551 : Vectors And Vector Operations

Example Question #552 : Vectors And Vector Operations

Example Question #553 : Vectors And Vector Operations

Example Question #554 : Vectors And Vector Operations

Example Question #555 : Vectors And Vector Operations

All Calculus 3 Resources

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