1.3 KiB
1.3 KiB
#rs/notes #rs/class/math163
- 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}
- Projection of one vector onto another:
- 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)
- equation:
- 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