20 lines
1.0 KiB
Markdown
20 lines
1.0 KiB
Markdown
#rs/notes #rs/class/math163
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- magnitude of the cross product
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- equation: $$||u \times v|| = ||u|| * ||v|| * \sin(\theta)$$
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- area of a parallelogram
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- if there are two (2D) vectors that are two adjacent sides, the area is $$||a \times b||$$
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- triple scalar product: $$u * (v \times w) = \begin{vmatrix} u_{1} && u_{2} && u_{3} \\ u_{1} && u_{2} && u_{3} \\ u_{1} && u_{2} && u_{3} \end{vmatrix}$$
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- (can either take the determinate of the matrix or the dot product of the cross product)
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- results in a scalar
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- This is equal to the volume of the parallelepiped with the three vectors representing adjacent edges
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- $$u * (v \times w) = (u \times v) * w $$
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- lines and planes in 3D space
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- can use 3D vectors to describe 3D lines
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- $PQ = tv$
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- $<x-x_{0}, y-y_{0},z-z_{0}> = t<a,b,c>$
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- $<x_{0}, y_{0}, z_{0>}$ is the initial point of the line (or just a point on the line)
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- $<a,b,c>$ is the 3D slope of the line
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- $t$ is the independent variable
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- can also write it as $$<x, y, z> = t<a,b,c> + <x_{0}, y_{0}, z_{0}>$$
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