ObsidianVault/Running Start/MATH&163 - Calculus 3/Class 10-7.md

#rs/notes #rs/class/math163

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• 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