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Calculus 3 : Normal Vectors

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Example Questions

Example Question #31 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 46\\ 12\\2 \end{bmatrix}$ 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, $\overrightarrow{a}=\begin{bmatrix} 46\\ 12\\2 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} -1\\4 \\1 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=-46+48+2=4$

The two vectors are not orthogonal.

Report an Error

Example Question #31 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 5\\ 3\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} -6\\-11 \end{bmatrix}$ , 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

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

Simply multiply terms across rows:

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

For our vectors, $\overrightarrow{a}=\begin{bmatrix} 5\\ 3\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} -6\\-11 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=-30-33=-63$

The two vectors are not orthogonal.

Report an Error

Example Question #33 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 4\\ 1\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} -3\\12 \end{bmatrix}$ , 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

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

Simply multiply terms across rows:

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

For our vectors, $\overrightarrow{a}=\begin{bmatrix} 4\\ 1\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} -3\\12 \end{bmatrix}$

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

The two vectors are orthogonal.

Report an Error

Example Question #34 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 5\\2 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}-10 \\ 4\end{bmatrix}$ , 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

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

Simply multiply terms across rows:

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

For our vectors, $\overrightarrow{a}=\begin{bmatrix} 5\\2 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}-10 \\ 4\end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=-50+4=-46$

The two vectors are not orthogonal.

Report an Error

Example Question #31 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 3\\ 7\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 14\\ -6\end{bmatrix}$ , 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

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

Simply multiply terms across rows:

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

For our vectors, $\overrightarrow{a}=\begin{bmatrix} 3\\ 7\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 14\\ -6\end{bmatrix}$

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

The two vectors are orthogonal.

Report an Error

Example Question #32 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 3\\2 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 12\\8 \end{bmatrix}$ , 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

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

Simply multiply terms across rows:

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

For our vectors, $\overrightarrow{a}=\begin{bmatrix} 3\\2 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 12\\8 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=36+16=52$

The two vectors are not orthogonal.

Report an Error

Example Question #36 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} 5\\ -2\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 8\\20 \end{bmatrix}$ , 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

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

Simply multiply terms across rows:

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

For our vectors, $\overrightarrow{a}=\begin{bmatrix} 5\\ -2\end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 8\\20 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=40-40=0$

The two vectors are orthogonal.

Report an Error

Example Question #33 : Normal Vectors

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

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

Simply multiply terms across rows:

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

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

$\overrightarrow{a}\cdot\overrightarrow{b}=22-11=11$

The two vectors are not orthogonal.

Report an Error

Example Question #34 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix}cos(x) \\sin(x) \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} cos(x)\\sin(x) \end{bmatrix}$ , are orthogonal or not.

Possible Answers:

The two vectors are not orthogonal.

The two vectors are orthogonal.

It cannot be determined unless is known.

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

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

Simply multiply terms across rows:

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

For our vectors, $\overrightarrow{a}=\begin{bmatrix}cos(x) \\sin(x) \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} cos(x)\\sin(x) \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=cos^2(x)+sin^2(x)=1$

The two vectors are not orthogonal.

Report an Error

Example Question #35 : Normal Vectors

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix} -10\\9 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 4.5\\5 \end{bmatrix}$ , 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

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

Simply multiply terms across rows:

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

For our vectors, $\overrightarrow{a}=\begin{bmatrix} -10\\9 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix} 4.5\\5 \end{bmatrix}$

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

The two vectors are orthogonal.

Report an Error

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Calculus 3 : Normal Vectors

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #31 : Normal Vectors

Example Question #31 : Normal Vectors

Example Question #33 : Normal Vectors

Example Question #34 : Normal Vectors

Example Question #31 : Normal Vectors

Example Question #32 : Normal Vectors

Example Question #36 : Normal Vectors

Example Question #33 : Normal Vectors

Example Question #34 : Normal Vectors

Example Question #35 : Normal Vectors

All Calculus 3 Resources

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