Dot Product
The dot product or inner product of \(\vec{v}= \begin{pmatrix} v_1 \\\ v_2 \end{pmatrix}\) and \(\vec{w}= \begin{pmatrix} w_1 \\\ w_2 \end{pmatrix}\) is
\[\vec{v} \cdot \vec{w} = v_1w_1 + v_2w_2\]Perpendicular
If \(\vec{v} \cdot \vec{w}=0\), \(\vec{v}\) and \(\vec{w}\) are perpendicular.
Length
The length \(\lvert \lvert \vec{v} \rvert \rvert\) of a vector \(\vec{v}\) is
\[\lvert \lvert \vec{v} \rvert \rvert = \sqrt{\vec{v} \cdot \vec{v}}\]Unit Vector
A unit vector \(\vec{u}\) is a vector whose length is 1.
\[\lvert \lvert \vec{u} \rvert \rvert = 1\]To any non-zero vector \(\vec{v}\), divide by its length to obtain a unit vector in the same direction as \(\vec{v}\)
\[\vec{u} = \frac{\vec{v}}{\lvert \lvert \vec{v} \rvert \rvert}\]Standard Unit Vector
The standard unit vectors along x-axis and y-axis are \(\hat{i}\) and \(\hat{j}\).
Angle between vectors
The angle betweeen \(\vec{v}\) and \(\vec{w}\) is,
\[\cos \theta = \frac{\vec{v}}{\lvert \lvert \vec{v} \rvert \rvert} \frac{\vec{w}}{\lvert \lvert \vec{w} \rvert \rvert}\] \[= \vec{u} \cdot \vec{U}\]where \(\vec{u}\) and \(\vec{U}\) are the unit vectors of \(\vec{v}\) and \(\vec{w}\).
We can see that the two vectors are perpendicular or \(\theta = 90\) when \(\vec{v} \cdot \vec{w}=0\)
\[\lvert \vec{v} \cdot \vec{w} \rvert \leq \lvert \lvert \vec{v} \rvert \rvert + \lvert \lvert \vec{w} \rvert \rvert\]Schwarz Inquality
\[\lvert \lvert \vec{v} + \vec{w} \rvert \rvert \leq \lvert \lvert \vec{v} \rvert \rvert + \lvert \lvert \vec{w} \rvert \rvert\]Triangle Inequality
References
[1] Introduction to Linear Algebra, 5th Edition