The cross product

A vector product that gives a vector

The cross product is a way to multiply two vectors and get another vector. It is written

ec{A} imesec{B}

Unlike the dot product, which gives a scalar, the cross product gives a vector. The result is perpendicular to both original vectors.

The magnitude of the cross product is

|ec{A} imesec{B}| = |ec{A}||ec{B}|sin heta

where $heta$ is the angle between the vectors.

Why sine appears

The sine factor measures how much of one vector is perpendicular to the other. If two vectors point in the same direction, $heta = 0$ and $sin 0 = 0$, so the cross product is zero. If they are perpendicular, $heta = 90^circ$ and $sin 90^circ = 1$, so the magnitude is maximum.

This makes the cross product useful for rotational effects. Rotation is strongest when a force is applied perpendicular to a lever arm.

Direction and the right-hand rule

The direction of ec{A} imesec{B} is perpendicular to the plane containing ec{A} and ec{B}. There are two possible perpendicular directions, so we need a rule. The standard is the right-hand rule.

Point the fingers of your right hand in the direction of ec{A}. Curl them toward ec{B}. Your thumb points in the direction of ec{A} imesec{B}.

Order matters. In general,

ec{A} imesec{B} = -ec{B} imesec{A}

Reversing the order reverses the direction.

Torque

A major physics use of the cross product is torque. Torque measures the tendency of a force to cause rotation. It is defined as

ec{ au} = ec{r} imesec{F}

Here ec{r} is the position vector from the rotation axis to the point where the force is applied, and ec{F} is the force.

The magnitude is

$au = rFsin heta$

A door is easier to open when you push far from the hinge and perpendicular to the door. This gives a large $r$ and a large sine factor. Pushing near the hinge or directly toward the hinge produces little torque.

Angular momentum

Another important cross product is angular momentum:

ec{L} = ec{r} imesec{p}

where ec{p} is linear momentum. Angular momentum describes rotational motion. Planets orbiting the Sun, spinning tops, rotating wheels, and atoms all involve angular momentum.

Component form

In advanced work, the cross product can be computed from components. For vectors

angle$$ and $$ec{B} = langle B_x,B_y,B_z angle$$ the result is $$ec{A} imesec{B} = langle A_yB_z-A_zB_y, A_zB_x-A_xB_z, A_xB_y-A_yB_x angle$$ This formula looks complicated, but its meaning remains geometric: magnitude from perpendicularity, direction from the right-hand rule. ## Cross product versus dot product The dot product measures parallel alignment and gives a scalar. The cross product measures perpendicular effect and gives a vector. Work uses the dot product because only force along displacement matters. Torque uses the cross product because only force perpendicular to the lever arm causes rotation. ## The big idea The cross product is the mathematics of perpendicular influence and rotation. It produces a vector perpendicular to two input vectors. Its size depends on $sin heta$, and its direction follows the right-hand rule. It is essential for torque, angular momentum, magnetic forces, and rotational physics.