## Volume ![[../../00 Excalidraw/Parallelepiped .excalidraw]] Base: $\norm{\vec u \times \vec v}$ Height: $\norm{\vec w} \sin\theta$ Find the angle between $\vec w$ and the planed created with $\vec u$ and $\vec v$ ([[Planes#Getting the angle between a plane and a line|see also]]). $\huge\begin{align*} \vec u \times \vec v &= \vec n\\ \sin \theta &= \frac{\vec n \cdot \vec w}{\norms{\vec n}\norms{\vec w}} \\ \text{height} &= \frac{\vec n \cdot \vec w}{\norms{\vec n}\norms{\vec w}}\\ \end{align*}$ > [!definition] The area of a parallelepiped is the [[Triple Product]] between it's defining vectors: $\huge \paren{ \vec u \times \vec v} \cdot \vec w $ >[!example] >Let $\vec u$, $\vec v$, and $\vec w$ be non coplanar and describing a parallelepiped, what is the volume of the shape? > $\huge\begin{align*} \vec u &= \mat{1\\-2\\4} \\ \vec v &= \mat{-1\\0\\2} \\ \vec w &= \mat{-1\\-1\\3} \\ (\vec u \times \vec v) \cdot \vec w &= \mat{ -4-0\\ -4-2\\ 0-2 } \cdot \mat{-1\\-1\\3}\\ &= 4 + 6 -6\\ &= 4 \end{align*}$ ### Related shapes >[!definition] Area of a Triangular Prism Similar to [[Parallelogram#Finding the area of a triangle|triangles]] in $\R^2$, it is simply half the the parallelepiped $\huge \frac{1}{2}\norm{\paren{\vec u \times \vec v} \cdot \vec w}$ >[!definition] Area of a pyramid Just accept it. $\huge \frac{1}{3}\norm{\paren{\vec u \times \vec v} \cdot \vec w}$ >[!definition] Area of a tetrahedron $\huge \frac{1}{6}\norm{\paren{\vec u \times \vec v} \cdot \vec w}$