In the graph below, move the points A, B or C and observe the change to the respestive vector, together with the change in the volume.
Wednesday, May 20, 2020
Volume of Parallelepiped
We already knew that the volume of a box = area x height, but a slanted box (parallelepiped) like the picture below? We may still use the concept of area x height to find the volume. But how? First, assign three vectors u, v and w that start from the same point as in the graph below. We can see that points A and B are both fall on the xy-plane. Considering that t could be an obtuse angle, absolute sign is put in when calculating the height. As such, we get
\cdot\textbf{w}|\end{align*} "\begin{align*} \textrm{Area} &=|\textbf{u}\times \textbf{v}|\\ \textrm{Height}&=|\textbf{w}||\cos t|\\ \textrm{Volume}&=|\textbf{u}\times \textbf{v}||\textbf{w}||\cos t|\\ &=|(\textbf{u}\times \textbf{v})\cdot\textbf{w}|\end{align*}")
\cdot\textbf{w} "(\textbf{u}\times \textbf{v})\cdot\textbf{w}")
is also named as the box product. Besides finding the cross product, followed by the dot product, you may combine the three vectors into a 3-by-3 matrix and use the determinant method to find the box product (refer in the box below).
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