Definition

A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. A unit vector is a vector with magnitude 1 and any argument value. A vector is denoted as $\overline{u}$ or $\overrightarrow{u}$.

Dot Product

$\overrightarrow{u} \cdot \overrightarrow{v} = u_1v_1+u_2v_2 + ... + u_nv_n = |\overrightarrow{u}||\overrightarrow{v}|\cos\theta$

• dot product only works with two vectors

Properties

If $\overrightarrow{u} \cdot \overrightarrow{v} =$

• $0$, $\overrightarrow{u} \perp \overrightarrow{v}$
• $< 0$, argument/angle is obtuse
• $> 0$, argument/angle is acute
• $1$, the two vectors are parallel
• $-1$, parallel and in opposite directions

Projection

$proj_{\overrightarrow{u}}^{\overrightarrow{v}} = \frac{\overrightarrow{u} \cdot \overrightarrow{v}}{\overrightarrow{u} \cdot \overrightarrow{v}}\overrightarrow{u}$

Cross Product

• $||\overline{v} \times \overline{w}|| = |\overline{v}||\overline{w}|\sin\theta$
• $\overline{u} \times \overline{v} = -\overline{v} \times \overline{u}$
• $\overline{u} \times \overline{v} \perp \overline{u}$ and $\overline{v}$

Parallelepiped

• volume = $|\overline{u}\cdot(\overline{v}\times\overline{w})|$

Distance

• point to plane $\Rightarrow ||proj_{\overline{p}}\overrightarrow{PQ}|| =$ $|\frac{\overrightarrow{PQ}\cdot\overline{p}}{||\overline{p}||}$