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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 #13 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix}2 \\0 \\5 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}10 \\7 \\-2 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix}2 \\0 \\5 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}10 \\7 \\-2 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=20+0-10=10$

The two vectors are not orthogonal.

Report an Error

Example Question #14 : Normal Vectors

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

Possible Answers:

The two vectors are orthogonal.

The two vectors are not orthogonal.

Correct answer:

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

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

The two vectors are orthogonal.

Report an Error

Example Question #15 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix}7 \\1 \\3 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}0 \\-5 \\ 2\end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix}7 \\1 \\3 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}0 \\-5 \\ 2\end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=0-5+6=1$

The two vectors are not orthogonal.

Report an Error

Example Question #16 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 3\\7 \\ 5\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 3\\3 \\-6 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} 3\\7 \\ 5\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 3\\3 \\-6 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=9+21-30=0$

The two vectors are orthogonal.

Report an Error

Example Question #21 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 0\\3 \\-5 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 4\\15 \\9 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} 0\\3 \\-5 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 4\\15 \\9 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=0+45-45=0$

The two vectors are orthogonal.

Report an Error

Example Question #22 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 4\\6 \\ -1\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}8 \\-5 \\1 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} 4\\6 \\ -1\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}8 \\-5 \\1 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=32-30-1=1$

The two vectors are not orthogonal.

Report an Error

Example Question #23 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 10\\-3 \\4 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 7\\ 10\\10 \end{bmatrix}$ , are orthogonal or not.

Possible Answers:

The two vectors are not orthogonal.

The two vectors are orthogonal.

Correct answer:

The two vectors are 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, $\overrightarrow{a}=\begin{bmatrix} 10\\-3 \\4 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 7\\ 10\\10 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=70-30+40=80$

The two vectors are not orthogonal.

Report an Error

Example Question #24 : Normal Vectors

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

Possible Answers:

The two vectors are orthogonal.

The two vectors are not orthogonal.

Correct answer:

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

$\overrightarrow{a}\cdot\overrightarrow{b}=-1-3+4=0$

The two vectors are orthogonal.

Report an Error

Example Question #25 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 6\\3 \\1 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}2 \\ 4\\0 \end{bmatrix}$ , 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, $\overrightarrow{a}=\begin{bmatrix} 6\\3 \\1 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}2 \\ 4\\0 \end{bmatrix}$

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

The two vectors are not orthogonal.

Report an Error

Example Question #2521 : Calculus 3

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

$\overrightarrow{a}\cdot\overrightarrow{b}=7+1+6=14$

The two vectors are not 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 #13 : Normal Vectors

Example Question #14 : Normal Vectors

Example Question #15 : Normal Vectors

Example Question #16 : Normal Vectors

Example Question #21 : Normal Vectors

Example Question #22 : Normal Vectors

Example Question #23 : Normal Vectors

Example Question #24 : Normal Vectors

Example Question #25 : Normal Vectors

Example Question #2521 : Calculus 3

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