Here is a parallelepiped.I want to determine the volume of the parallelepiped.
One of my friends said to me that the volume of the parallelepiped can be found out by the following formula.
$${\rm Volume}=|\vec a\cdot(\vec b\times \vec c)|$$
I can not understand why and how this formula works? Can anyone explain with clarification.
Best Answer
Think of it this way: Area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is given by $|\vec{b} \times \vec{c}|$. This is easy to see because the area of a parallelogram is $base \times height = c b \sin \theta $ where $\theta$ is the angle between $\vec{b}$ and $\vec{c}$.
In order to get the volume, we need to multiply this area by the projection of $\vec{a}$ along the direction perpendicular to the parallelogram formed by $\vec{b}$ and $\vec{c}$ which is given by the formula you have mentioned because $ \vec{b} \times \vec{c} $ is a vector perpendicular to the plane (the idea is the same - $base \times height$ - the base is now an area rather than a length)