Tags: [[Calculus 3]]
- Direction angles
	- angles vector forms with axis
- vector projection
	- Projection of one vector onto another: $$proj_{u}v=\frac{u\cdot v}{\left|\left|u\right|\right|^{2}}u$$
	- Vector that is same direction
	- To just get the magnitude of the projected vector, use: $$ mag_{u}v=\frac{|u*v|}{||u||^2} $$
- Unit vectors
	- magnitude is one
	- Formula: $$ unit_u=\frac{u}{||u||}$$
- Resolving vectors to components
	- project one vector to another
	- subtract projection from original vector
- Determinate
	- equation: $$\begin{vmatrix} a&b\\c&d\end{vmatrix}\rightarrow ad-bc$$
	- larger than 2x2: $$\begin{vmatrix} a&b&c\\d&e&f\\g&h&i\end{vmatrix}\rightarrow a\begin{vmatrix} e&f\\h&i\end{vmatrix} - b\begin{vmatrix} d&f\\g&i\end{vmatrix} + c\begin{vmatrix} d&e\\g&h\end{vmatrix}\rightarrow a(ei-hf)-b(di-gf)+c(dh-ge)$$
- Cross product
	- creates vector that is orthogonal to both vectors
	- for which direction it goes, use right hand rule
		- pointer finger is first vector, middle is second
		- thumb is resulting vector
	- equation: $$a \times b = \begin{vmatrix} i&j&k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}= i(a_2b_3-a_3b_2)-j(a_1b_3-a_3b_1)+k(a_1b_2-a_2b_1)$$
	- cross product is not communitive $$u\times v \ne v \times u$$
	- it is anti-communitive though $$ u \times v = -(v \times u) $$
	- this too $$ a \times a = 0 $$