Vectors, Dot Products and Cross Products

Two products answer different questions

A vector in space has three ordered components. Its length is $\|v\|=\sqrt{v_x^2+v_y^2+v_z^2}$ . The dot product $u\cdot v=u_xv_x+u_yv_y+u_zv_z$ is a scalar. For nonzero vectors it equals $\|u\|\|v\|\cos\theta$ , measuring alignment. Orthogonality means the dot product is zero.

The cross product is a vector perpendicular to both inputs:

u\times v=(u_yv_z-u_zv_y,\ u_zv_x-u_xv_z,\ u_xv_y-u_yv_x).

Its magnitude is the area of the parallelogram spanned by the inputs. Its direction follows the right-hand rule, so swapping inputs reverses its sign. Parallel inputs, including a zero input, give a zero cross product and no resulting normal direction.

Visual guide

VISUAL GUIDEDot product keeps only the parallel part

Perpendicular component

For displacement d = (2, 0) and force F = (3, 4), only the horizontal component 3 contributes to work: F · d = 6. The vertical component is perpendicular to the displacement.

Worked example: force and displacement

For force $F=(3,4,0)$ N and displacement $d=(2,0,0)$ m, work is $F\cdot d=6$ J. The perpendicular force component contributes no work to this displacement. The projection of $F$ onto the displacement direction is $(F\cdot d)/(d\cdot d)\,d=(3,0,0)$ N.

Do not replace work with $\|F\|\|d\|=10$ J; that assumes alignment. Units help distinguish a force vector, a displacement and their scalar work.

Worked example: oriented area and volume

For $u=(1,0,0)$ and $v=(0,2,0)$ , the cross product is $(0,0,2)$ , so the parallelogram area is $2$ and its upward normal is explicit. With $w=(0,0,3)$ , the scalar triple product $u\cdot(v\times w)=6$ is signed volume. Geometric volume is its absolute value.

A zero triple product indicates coplanar vectors. A negative value records reversed orientation, not negative physical volume. Torque instead uses $r\times F$ , where $r$ is measured from the specified origin; changing that origin generally changes torque.

Practice

Find the angle between $(1,1,0)$ and $(1,-1,0)$ .
Find the area of the triangle spanned by $(2,0,0)$ and $(0,3,0)$ .
Compute $(0,1,0)\times(1,0,0)$ .

Show worked solutions

Their dot product is zero and both are nonzero, so the angle is $\pi/2$ .
The parallelogram area is $6$ ; the triangle area is $3$ .
The result is $(0,0,-1)$ , opposite to the product in the reverse order.

Further study

MIT OpenCourseWare: Multivariable Calculus provides a full university course with additional lectures and exercises.