Linear algebra#
Linear combinations#
\[
\vec{0} \ne 0
\]
Problem 1.1A#
Describe the plane the vectors \(\vec{v} = (1,1,0)\) and \(\vec{w} = (0,1,1)\) fill.
\[
c\vec{v} + d\vec{w} = (c, c+d, d)
\]
Dot products and lengths#
\[\begin{split}
\vec{v}\cdot\vec{w} = v_1w_1 + v_2w_2 \\
\Vert\vec{v}\Vert = \sqrt{\vec{v}\cdot\vec{v}} = \sqrt{\sum v_i^2}
\end{split}\]
If
\[
\vec{v}\cdot\vec{w} = 0
\]
then \(\vec{v}\) ans \(\vec{w}\) are perpendicular.
The angle between \(\vec{v}\) and \(\vec{w}\) is given by:
\[
\cos\theta = \frac{\vec{v}\cdot\vec{w}}{\Vert\vec{v}\Vert\Vert\vec{w}\Vert}
\]