27 lines
1.3 KiB
Markdown
27 lines
1.3 KiB
Markdown
#rs/notes #rs/class/math163
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- - -
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- Direction angles
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- angles vector forms with axis
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- vector projection
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- Projection of one vector onto another: $$proj_{u}v=\frac{u\cdot v}{\left|\left|u\right|\right|^{2}}u$$
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- Vector that is same direction
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- To just get the magnitude of the projected vector, use: $$ mag_{u}v=\frac{|u*v|}{||u||^2} $$
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- Unit vectors
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- magnitude is one
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- Formula: $$ unit_u=\frac{u}{||u||}$$
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- Resolving vectors to components
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- project one vector to another
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- subtract projection from original vector
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- Determinate
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- equation: $$\begin{vmatrix} a&b\\c&d\end{vmatrix}\rightarrow ad-bc$$
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- larger than 2x2: $$\begin{vmatrix} a&b&c\\d&e&f\\g&h&i\end{vmatrix}\rightarrow a\begin{vmatrix} e&f\\h&i\end{vmatrix} - b\begin{vmatrix} d&f\\g&i\end{vmatrix} + c\begin{vmatrix} d&e\\g&h\end{vmatrix}\rightarrow a(ei-hf)-b(di-gf)+c(dh-ge)$$
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- Cross product
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- creates vector that is orthogonal to both vectors
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- for which direction it goes, use right hand rule
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- pointer finger is first vector, middle is second
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- thumb is resulting vector
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- equation: $$a \times b = \begin{vmatrix} i&j&k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}= i(a_2b_3-a_3b_2)-j(a_1b_3-a_3b_1)+k(a_1b_2-a_2b_1)$$
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- cross product is not communitive $$u\times v \ne v \times u$$
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- it is anti-communitive though $$ u \times v = -(v \times u) $$
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- this too $$ a \times a = 0 $$
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