Vector Multiplication

Two Kinds of Vector Multiplication

Vector multiplication does not always produce another vector. Two kinds of multiplication are common in physics: the dot product and the cross product.

The dot product produces a scalar. The cross product produces a new vector whose direction is perpendicular to the two original vectors.

Type of multiplication	Symbol	Result	Physics example
dot product	$\vec{A}\cdot\vec{B}$	scalar	work
cross product	$\vec{A}\times\vec{B}$	vector	torque

The multiplication sign changes because the physical question changes: how much is in the same direction, or how strong is the turning effect.

Dot Product Measures the Same-Direction Part

The dot product of $\vec{A}$ and $\vec{B}$ is defined as:

\vec{A}\cdot\vec{B}=AB\cos\theta

Angle $\theta$ is the angle between the two vectors. The result is a scalar because it gives a value, not a direction.

In physics, work done by a force can be written as:

W=\vec{F}\cdot\vec{s}=Fs\cos\theta

This means only the component of force along the displacement actually does work.

Work from Dot Product

A person pulls a box with a $40\text{ N}$ force while the box moves $5\text{ m}$ . The force makes a $60^\circ$ angle with the displacement direction. The work is:

\begin{aligned} W&=Fs\cos\theta \\ &=40(5)\cos60^\circ \\ &=200(0.5) \\ &=100\text{ J} \end{aligned}

If the force is perpendicular to the displacement, $\theta=90^\circ$ and $\cos90^\circ=0$ . In that case, the force does no work in the displacement direction.

Cross Product Measures Turning Effect

The cross product of $\vec{A}$ and $\vec{B}$ has magnitude:

\lvert\vec{A}\times\vec{B}\rvert=AB\sin\theta

The result is a vector perpendicular to the plane containing $\vec{A}$ and $\vec{B}$ . Its direction is determined by the right-hand rule.

Direction of

\vec{A}\times\vec{B}

Vectors

\vec{A}

and

\vec{B}

lie in one plane. Their cross product points perpendicular to that plane.

In this visual, $\vec{A}$ points along the positive $x$ axis and $\vec{B}$ points along the positive $y$ axis. The cross-product formula gives the positive $z$ direction for $\vec{A}\times\vec{B}$ . The blue arrow is scaled down to fit the scene, but its direction still comes from the calculation.

For torque, the cross product is written:

\vec{\tau}=\vec{r}\times\vec{F}

Vector $\vec{r}$ is the lever arm from the rotation axis to the point where the force acts. The same force produces a larger torque when applied farther from a door hinge.

Torque from Perpendicular Force

A door is pushed with a $30\text{ N}$ force at a distance of $0.8\text{ m}$ from the hinge. The force is perpendicular to the door, so $\theta=90^\circ$ .

\begin{aligned} \tau&=rF\sin\theta \\ &=0.8(30)\sin90^\circ \\ &=24\text{ N m} \end{aligned}

If the force is applied at a smaller angle, $\sin\theta$ is also smaller. That is why pushing a door perpendicularly is usually more effective than pushing almost parallel to the door surface.

Comparing Dot and Cross Products

The main difference can be seen from the angle. The dot product is largest when two vectors point the same way because $\cos0^\circ=1$ . The cross product is zero in that same condition because $\sin0^\circ=0$ .

Angle condition	$\vec{A}\cdot\vec{B}$	$\lvert\vec{A}\times\vec{B}\rvert$
$0^\circ$	maximum	zero
$90^\circ$	zero	maximum
$180^\circ$	negative	zero

So the dot product is useful for asking about alignment, while the cross product is useful for perpendicular effects such as rotation.