Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}\)

To find the dot product of two vectors given the notation

\(\displaystyle \overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}\)

Simply multiply terms across rows:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3\)

For our vectors, \(\displaystyle \overrightarrow{a}=\begin{bmatrix}9 \\16 \\25 \end{bmatrix}\) and \(\displaystyle \overrightarrow{b}=\begin{bmatrix} 9\\ 16\\-25 \end{bmatrix}\)

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=81+256-625=-288\)

The two vectors are not orthogonal.

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}\)

To find the dot product of two vectors given the notation

\(\displaystyle \overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}\)

Simply multiply terms across rows:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3\)

For our vectors, \(\displaystyle \overrightarrow{a}=\begin{bmatrix} 9\\-5 \\4 \end{bmatrix}\) and \(\displaystyle \overrightarrow{b}=\begin{bmatrix} 2\\ 2\\ -2\end{bmatrix}\)

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=18-10-8=0\)

The two vectors are orthogonal.

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}\)

To find the dot product of two vectors given the notation

\(\displaystyle \overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}\)

Simply multiply terms across rows:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3\)

For our vectors, \(\displaystyle \overrightarrow{a}=\begin{bmatrix} 8\\ -7\\ 3\end{bmatrix}\) and \(\displaystyle \overrightarrow{b}=\begin{bmatrix} 4\\1 \\2 \end{bmatrix}\)

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=32-7+6=31\)

The two vectors are not orthogonal.

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}\)

To find the dot product of two vectors given the notation

\(\displaystyle \overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}\)

Simply multiply terms across rows:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3\)

For our vectors, \(\displaystyle \overrightarrow{a}=\begin{bmatrix}3 \\2 \\-9 \end{bmatrix}\) and \(\displaystyle \overrightarrow{b}=\begin{bmatrix} 4\\3 \\2 \end{bmatrix}\)

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=12+6-18=0\)

The two vectors are orthogonal.

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}\)

To find the dot product of two vectors given the notation

\(\displaystyle \overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}\)

Simply multiply terms across rows:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3\)

For our vectors, \(\displaystyle \overrightarrow{a}=\begin{bmatrix} 1\\ 9\\-9 \end{bmatrix}\) and \(\displaystyle \overrightarrow{b}=\begin{bmatrix} 1\\1 \\1 \end{bmatrix}\)

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=1+9-9=1\)

The two vectors are not orthogonal.

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}\)

To find the dot product of two vectors given the notation

\(\displaystyle \overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}\)

Simply multiply terms across rows:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3\)

For our vectors, \(\displaystyle \overrightarrow{a}=\begin{bmatrix} 2\\-6 \\-7 \end{bmatrix}\) and \(\displaystyle \overrightarrow{b}=\begin{bmatrix}3 \\-6 \\6 \end{bmatrix}\)

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=6+36-42=0\)

The two vectors are orthogonal.

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}\)

To find the dot product of two vectors given the notation

\(\displaystyle \overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}\)

Simply multiply terms across rows:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3\)

For our vectors, \(\displaystyle \overrightarrow{a}=\begin{bmatrix} -4\\-6 \\-7 \end{bmatrix}\) and \(\displaystyle \overrightarrow{b}=\begin{bmatrix} 1\\-5 \\4 \end{bmatrix}\)

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=-4+30-28=-2\)

The two vectors are not orthogonal.

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}\)

To find the dot product of two vectors given the notation

\(\displaystyle \overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}\)

Simply multiply terms across rows:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3\)

For our vectors, \(\displaystyle \overrightarrow{a}=\begin{bmatrix} 6\\ -6\\5 \end{bmatrix}\) and \(\displaystyle \overrightarrow{b}=\begin{bmatrix} 13\\ -2\\ -18\end{bmatrix}\)

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=78+12-90=0\)

The two vectors are orthogonal.

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}\)

To find the dot product of two vectors given the notation

\(\displaystyle \overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}\)

Simply multiply terms across rows:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3\)

For our vectors, \(\displaystyle \overrightarrow{a}=\begin{bmatrix} -1\\5 \\8 \end{bmatrix}\) and \(\displaystyle \overrightarrow{b}=\begin{bmatrix} 3\\4 \\ -3\end{bmatrix}\)

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=-3+20-24=-7\)

The two vectors are not orthogonal.

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}\)

To find the dot product of two vectors given the notation

\(\displaystyle \overrightarrow{a}\begin{bmatrix} a_1\\a_2 \\ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\\b_2 \\ b_3 \end{bmatrix}\)

Simply multiply terms across rows:

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3\)

For our vectors, \(\displaystyle \overrightarrow{a}=\begin{bmatrix}1 \\ -9\\1 \end{bmatrix}\) and \(\displaystyle \overrightarrow{b}=\begin{bmatrix} 13\\1 \\-4 \end{bmatrix}\)

\(\displaystyle \overrightarrow{a}\cdot\overrightarrow{b}=13-9-4=0\)

The two vectors are orthogonal.