Work

Work is the force applied onto an object over a distance.

You can calculate the work done on an object by using the dot product:

Work $=$ magnitude of a force $\vec {F}$ along $\vec {d} \ \times$ magnitude of displacement

$\quad \quad \ = | \ \text {proj}_{ \vec {d} } \vec {F} \ | \times | \ \vec {d} \ |$

$\quad \quad \ = \frac {| \ \vec {F} \ \cdot \ \vec {d} \ |} {| \ \vec {d} \ |} \times | \ \vec {d} \ |$

$\quad \quad \ = | \ \vec {F} \ \cdot \ \vec {d} \ |$

So, the work for a force $\vec {F}$ along a distance $\vec {d}$ is:

$\text {Work} = | \ \vec {F} \ \cdot \ \vec {d} \ |$

Torque

Torque is a vector quantity, denoted as $\vec { \tau }$, is the measure of force applied about an axis in order to rotate an object.

Torque can be calculated using the cross product of the radius $\vec {r}$ from the object, and the force $\vec {F}$ applied:

$| \ \vec {\tau} \ | = | \ \vec {r} \times \vec {F} \ | = | \ \vec {r} \ | | \ \vec {F} \ | \sin \theta$

Area of a parallelogram

The area of a parallelogram formed by an upper vector $\vec {a}$ and a lower vector $\vec {b}$ is given by the magnitude of the cross product of both vectors:

$A = | \ \vec {a} \times \vec {b} \ | = | \ \vec {a} \ | | \ \vec {b} \ | \sin \theta$

Volume of a parallelepiped

The volume of a parallelepiped formed by a vector $\vec {c}$ and two lower vectors $\vec {a}$ and $\vec {b}$ is given by the magnitude of the cross product of both vectors:

$V = | \ \vec {a}\ \cdot \ (\vec {b} \times \vec {c}) \ |$