Cross Product

Vector

• Definition
  • Cross Product is a vector
  • How to compute
• Theorem
  • Length: $|\vec{A}\times \vec{B}|=|\vec{A}||\vec{B}|\sin \theta$ the area of the parallelogram they formed in space
    • direction of ($\vec{A}\times \vec{B}$) is perpendicular to the plane
  • Right-hand-rule
    • index finger change $[0,\pi ]$ and thumbs up
  • Volume = area(base) x height = $|\vec{A}\times \vec{B}|\cdot \vec{A}\cdot \vec{n}$ = $\vec{A}\cdot (\vec{B}\times \vec{C})$ = determinant , where $\vec{n}=\frac{\vec{B}\times \vec{C}}{|\vec{B}\times \vec{C}|}$ , a unit vector, the component of $\vec{A}$ in vertical direction. aka triple product
  • $\vec{A}\times \vec{B}=-\vec{B}\times \vec{A}$
  • $\vec{A}\times \vec{A}=0$
• Application
  • The equation of the plane
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    • Find normal vector (using cross product), find P(x, y, z) , get $\overrightarrow{P{P}_1}$ , then $\overrightarrow{P{P}_1}\cdot \vec{N}$ will return the equation.

Numberic