The vector (cross) product

The second kind of vector product — and the more surprising one. Unlike the dot product (which yields a scalar), the cross product yields a vector — with direction fixed by the right-hand rule. Needed for torque, angular momentum, Lorentz force, and everything rotational.

Definition

For two vectors \vec a and \vec b with angle \phi between them:

$$
\boxed{\vec a \times \vec b = |a||b|\sin\phi\ \hat n}
$$

where:
– magnitude: |a||b|\sin\phi
– direction: \hat n — the unit vector perpendicular to the plane containing \vec a and \vec b
– sign of \hat n: given by the right-hand rule

The right-hand rule

With your right hand:
– Point fingers along \vec a (first vector)
– Curl them toward \vec b (second) — via the shorter angle
– Your thumb points along \vec a \times \vec b

Reversing the order (\vec b \times \vec a) reverses the thumb.

Key difference — anti-commutative

$$
\vec a \times \vec b = -(\vec b \times \vec a)
$$

Order matters! Unlike the dot product (\vec a \cdot \vec b = \vec b \cdot \vec a), the cross product has the same magnitude but opposite sign when reversed.

Signs and special cases

• \phi = 0° or 180°: \sin\phi = 0 → \vec a \times \vec b = 0 (parallel vectors have zero cross product)
• \phi = 90°: \sin\phi = 1 → maximum magnitude: |a||b|

Important: if two vectors are parallel (same direction or opposite), their cross product is zero.

Component form — the 3×3 determinant

Starting from unit vectors:

• \hat i \times \hat j = \hat k
• \hat j \times \hat k = \hat i
• \hat k \times \hat i = \hat j
• Reverse gives minus: \hat j \times \hat i = -\hat k, etc.

Cyclic rule: i → j → k → i → .... Forward is positive, backward is negative.

Full formula:

$$
\vec a \times \vec b = \begin{vmatrix} \hat i & \hat j & \hat k \ a_x & a_y & a_z \ b_x & b_y & b_z \end{vmatrix}
$$

Expanded:

$$
\vec a \times \vec b = (a_y b_z – a_z b_y)\hat i + (a_z b_x – a_x b_z)\hat j + (a_x b_y – a_y b_x)\hat k
$$

Numerical example

Scenario: \vec a = 2\hat i + 3\hat j + \hat k, \vec b = \hat i - \hat j + 2\hat k. Compute \vec a \times \vec b.

x component: a_y b_z - a_z b_y = 3(2) - 1(-1) = 6 + 1 = 7

y component: a_z b_x - a_x b_z = 1(1) - 2(2) = 1 - 4 = -3

z component: a_x b_y - a_y b_x = 2(-1) - 3(1) = -2 - 3 = -5

$$
\vec a \times \vec b = 7\hat i – 3\hat j – 5\hat k
$$

Sanity check: the dot product of this result with either \vec a or \vec b must be zero (since the cross product is perpendicular to both).

$$
(\vec a \times \vec b) \cdot \vec a = 7(2) + (-3)(3) + (-5)(1) = 14 – 9 – 5 = 0\ ✓
$$

Physical application 1 — torque

Torque is a force acting about an axis:

$$
\vec \tau = \vec r \times \vec F
$$

where \vec r is the position vector (axis to point of application) and \vec F the force.

Interpretation: only the force component perpendicular to \vec r produces torque. Parallel component contributes nothing.

Everyday example: a wrench. Push perpendicular to the handle for maximum torque. Push along the handle — zero torque (just pulling, not twisting).

Physical application 2 — angular momentum

$$
\vec L = \vec r \times \vec p
$$

where \vec p = m\vec v is linear momentum. For a particle in orbit, \vec L is perpendicular to the orbital plane — and is conserved under central forces.

Physical application 3 — magnetic force

$$
\vec F = q\vec v \times \vec B
$$

Force on a charge q moving with velocity \vec v in a magnetic field \vec B. Always perpendicular to velocity — so it changes direction only, not speed. Result: circular orbit in a uniform magnetic field.

A few notes and common mistakes

1. Use the right hand, not the left.
Most common mistake. Practice with the right hand. If you’re left-handed as a physicist, still use “right hand” — not your dominant hand.

2. Parallel vectors give zero cross product.
\vec a \times \vec a = 0 always. \vec a \times (2\vec a) = 0 too.

3. Cross product is not commutative.
\vec a \times \vec b \neq \vec b \times \vec a. Watch the sign.

4. Magnitude = parallelogram area.
Geometric interpretation: |a||b|\sin\phi is the area of the parallelogram with the two vectors as sides. Triangle: half that.

5. Cross product exists only in 3D.
In 2D, the “plane” of two vectors is unique — no sense of a cross-product direction. (Abstract generalization to higher dimensions (“wedge product”) exists but is beyond this chapter.)

What you should be able to do

After this section, you should be able to:

• Write the cross-product definition |a||b|\sin\phi\ \hat n
• Use the right-hand rule to fix direction
• Compute via the determinant form
• Compute torque as \vec r \times \vec F
• Recognize when the cross product is zero (parallel vectors)
• Notice anti-commutativity

Preview of §3.7

Seven worked problems exercising every Chapter 3 tool: decomposition, addition, dot product, cross product, work, torque. §3.7 is next.

📚 See also: Halliday Vol 1, Ch 3, §3.6 — Multiplying Vectors: Vector Product.