三重积、力矩与体积

The scalar triple product packages volume into one determinant

The sign of the triple product records orientation

Torque is the cross product of the lever arm with the force

Rotation of a rigid body brings the pieces together

Let $\mathbf{a} = (2, 0, 0)$, $\mathbf{b} = (1, 3, 0)$ and $\mathbf{c} = (0, 1, 4)$. What is the value of $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$?

Three edge vectors from one corner of a parallelepiped are $(1, 0, 0)$, $(0, 2, 0)$ and $(1, 1, 3)$. What is the volume of the solid they span?

A mechanic pulls a wrench with force $\mathbf{F} = (0, 20, 0)$ newtons at the point $\mathbf{r} = (0.5, 0, 0)$ meters from the bolt. What is the torque about the bolt?

Three vectors satisfy $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = -7$. What does the sign tell you about the ordered triple $(\mathbf{a}, \mathbf{b}, \mathbf{c})$?

Three edge vectors from one vertex of a tetrahedron are $(2, 0, 0)$, $(0, 3, 0)$ and $(0, 0, 6)$. The tetrahedron's volume is a fixed fraction of the parallelepiped volume. Give the tetrahedron volume as a number.

A parallelepiped has edge vectors $(1, 0, 0)$, $(0, 2, 0)$ and $(0, 1, h)$, and its volume is $4$. Give the value of $h$.