Dot product / scalar product: 0
$= \textbf{a} \cdot \textbf{b} = |\textbf{a}| |\textbf{b}| \cos\measuredangle(\textbf{a},\textbf{b}) = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \cdot \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} = a_1 b_1+a_2 b_2+a_3 b_3 $
Angle: 0
$= \measuredangle(\textbf{a},\textbf{b}) = \arccos\left( \frac{ \textbf{a}\cdot\textbf{b} } {\left|\textbf{a}\right| \left|\textbf{b}\right| }\right)$
Length of the
projection: 		0
$= \left|\textbf{a}_\textbf{b}\right| = \left|\frac{\textbf{a}\cdot\textbf{b} } {\left|\textbf{b}\right|^2} \textbf{b}\right| $

How to calculate the angle $ \alpha $ between the two vectors $ \mathbf a $ and $ \mathbf b $?

$ \alpha = \cos (\textbf{a}\cdot\textbf{b}) $
$ \alpha = \arccos\left( \frac{ \textbf{a}\cdot\textbf{b} }{\left|\textbf{a}\right| \left|\textbf{b}\right| }\right) $
$ \alpha = \cos\left( \frac{ \textbf{a}\cdot\textbf{b} }{\left|\textbf{a}\right| \left|\textbf{b}\right| }\right) $
$ \alpha = \arcsin\left( \frac{ \textbf{a}\cdot\textbf{b} }{\left|\textbf{a}\right| \left|\textbf{b}\right| }\right) $
$ \alpha = \sin(\mathbf a \cdot \mathbf b) $

What is the scalar product of $\begin{pmatrix}1.5\\2\\7\end{pmatrix}$ and $\begin{pmatrix}8\\4.5\\3\end{pmatrix}$ ?

What is the scalar product of two orthogonal vectors?

What is the scalar product of two parallel vectors? (length = 1)

When is the dot product negative?

The angle is greater than 90 degree
The angle is less than 90 degree
The vectors do not have equal lengths
The vectors are parallel to each other

How to calculate the length of a vector?

$ |\textbf{a}| = \textbf{a} \cdot \textbf{a }$
$ |\textbf{a}| = \sqrt{\textbf{a} \cdot \textbf{a}} $
$ |\textbf{a}| = \textbf{a} \cdot \textbf{a}^{-1} $
$ |\textbf{a}| = (\textbf{a} \cdot \textbf{a})^2$