1. **Problem Statement:** We want to understand how the equation for the height $h$ of a parallelepiped, given by $h = a \cdot \frac{b \times c}{|b \times c|}$, is derived. 2. **Background:** The volume $V$ of a parallelepiped with edges represented by vectors $a$, $b$, and $c$ is given by $$V = |a \cdot (b \times c)|$$ where $b \times c$ is the cross product of vectors $b$ and $c$. 3. **Key Concepts:** - The area of the base parallelogram formed by vectors $b$ and $c$ is $|b \times c|$. - The height $h$ is the perpendicular distance from the base to the top face, which is the component of vector $a$ perpendicular to the base. 4. **Deriving the Height Formula:** - The vector $b \times c$ is perpendicular to the plane containing $b$ and $c$. - To find the height, we project vector $a$ onto the direction of $b \times c$. - The unit vector perpendicular to the base is $$\hat{n} = \frac{b \times c}{|b \times c|}$$ - The height $h$ is the scalar projection of $a$ onto $\hat{n}$: $$h = a \cdot \hat{n} = a \cdot \frac{b \times c}{|b \times c|}$$ 5. **Putting it all together:** - Volume is base area times height: $$V = (\text{area of base})(\text{height}) = |b \times c| \times \left|a \cdot \frac{b \times c}{|b \times c|}\right| = |a \cdot (b \times c)|$$ 6. **Summary:** The height formula comes from projecting vector $a$ onto the normal vector of the base parallelogram, which is the unit vector in the direction of $b \times c$. This projection gives the perpendicular height needed to calculate the volume.