ObsidianVault/Running Start/MATH&163 - Calculus 3/Class 10-9.md

#rs/notes #rs/class/math163

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