Multiplying two vectors gives either a number or a new vector — depending on which product you use. This is not arbitrary. Work needs one, torque needs the other. Here is why.

Dot Product and Cross Product: Two Ways Directions Can Multiply

You know how to add vectors. Now the next question: can you multiply them?

The answer is yes — but with a surprise. There are two completely different ways to multiply two vectors, and they give two completely different kinds of answers. One gives a plain number. The other gives a new vector pointing in a direction neither original vector pointed.

This is not a mathematical accident. Physics needs both. Work is one kind of product. Torque is the other. Angular momentum is the other. Magnetic force is the other. Understanding which product to use, and why, is one of the most useful skills in all of physics.

The dot product: how much do they agree?

The dot product of two vectors A and B is written A · B and defined as:

$\mathbf{A} \cdot \mathbf{B} = AB\cos\theta$

where θ is the angle between them, and A and B are their magnitudes.

The result is a scalar — a plain number, no direction.

What does this number measure? It measures how much of A lies along the direction of B — or equivalently, how much of B lies along A. It is the product of the magnitudes, weighted by how parallel the two vectors are.

When θ = 0° (parallel): A · B = AB. Maximum. They are fully aligned. When θ = 90° (perpendicular): A · B = 0. Nothing in common. When θ = 180° (antiparallel): A · B = −AB. Minimum. Fully opposed.

The sign tells you whether the vectors lean toward each other or away.

Why work is a dot product

You push a box with force F at angle θ to the direction of motion d. The work done is:

$W = \mathbf{F} \cdot \mathbf{d} = Fd\cos\theta$

Why cosθ? Because only the component of force along the direction of motion does useful work. The perpendicular component pushes the box into the floor — it doesn't move the box forward. The dot product automatically extracts that parallel component.

Push horizontally, box moves horizontally: θ = 0°, W = Fd. Full work. Push perpendicular to motion (holding something up while it moves sideways): θ = 90°, W = 0. Zero work — no matter how hard you push. Friction force opposing motion: θ = 180°, W = −Fd. Negative work — energy taken from the box.

The dot product is not a formula someone invented for work. Work is a dot product — because that is precisely what the physics requires.

Dot product in component form

If A = Axî + Ayĵ + Azk̂ and B = Bxî + Byĵ + Bzk̂, then:

$\mathbf{A} \cdot \mathbf{B} = A_xB_x + A_yB_y + A_zB_z$

Multiply the matching components and add. No angles needed.

This follows from the fact that î · î = 1, ĵ · ĵ = 1, k̂ · k̂ = 1 (parallel unit vectors), and î · ĵ = 0, ĵ · k̂ = 0, k̂ · î = 0 (perpendicular unit vectors).

Note: Method 1 — Using angle: A · B = AB cosθ. Use when magnitudes and angle between vectors are given.

> > **Method 2 — Using components:** > A · B = AxBx + AyBy + AzBz. Use when vectors are given in component form. > > Both methods give the same answer. Choose whichever the problem makes easier.

Finding the angle between two vectors

The dot product gives a clean way to find the angle between any two vectors:

$\cos\theta = \frac{\mathbf{A} \cdot \mathbf{B}}{AB} = \frac{A_xB_x + A_yB_y + A_zB_z}{AB}$

This is one of the most useful applications. Given two vectors in component form, find their dot product, divide by the product of magnitudes, take the inverse cosine.

The cross product: what is perpendicular to both?

The cross product of A and B is written A × B and defined as:

$|\mathbf{A} \times \mathbf{B}| = AB\sin\theta$

But unlike the dot product, this result is a vector — it has both magnitude and direction.

The magnitude is AB sinθ. The direction is perpendicular to both A and B — it points out of the plane containing them.

Which perpendicular direction? There are two (up and down from the plane). The right-hand rule decides: curl the fingers of your right hand from A toward B through the smaller angle. Your thumb points in the direction of A × B.

{/* DIAGRAM: Right hand rule — fingers curling from A to B, thumb pointing up */}

When θ = 0° (parallel): |A × B| = 0. No perpendicular direction exists when they point the same way. When θ = 90° (perpendicular): |A × B| = AB. Maximum — they define a plane most clearly. When θ = 180° (antiparallel): |A × B| = 0. Again no unique perpendicular.

Why torque is a cross product

A force F is applied at position r from a pivot. The torque is:

$\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$

Why a cross product? Because torque has both a magnitude and an axis of rotation — and the axis is perpendicular to both the position and the force.

Push a door at the hinge: r is tiny, torque is tiny — it barely rotates. Push a door at the edge: r is maximum, torque is maximum — it swings easily. Push the door directly toward the hinge (force parallel to r): sinθ = 0, torque = 0 — no rotation at all, no matter how hard you push.

The cross product captures all of this. The sin θ term means only the perpendicular component of force contributes to rotation. The direction of τ tells you the axis — clockwise or anticlockwise, which way the door swings.

The cross product is not commutative

This is the critical difference from the dot product.

$\mathbf{A} \times \mathbf{B} = -(\mathbf{B} \times \mathbf{A})$

Reversing the order reverses the direction. Curl your fingers from B to A instead of A to B — your thumb points the other way.

This makes physical sense. Torque r × F points in the direction of the resulting rotation axis. If you reverse the position and force vectors, the rotation reverses. The sign matters.

The dot product doesn't have this issue — A · B = B · A — because a number has no direction to reverse.

Cross product in component form

$\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$

Expanding:

$\mathbf{A} \times \mathbf{B} = (A_yB_z - A_zB_y)\hat{i} - (A_xB_z - A_zB_x)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$

The determinant form is the cleanest way to remember this. The pattern: for each component, multiply the other two in a cross pattern and subtract.

Note: Method 1 — Using angle: |A × B| = AB sinθ, direction from right-hand rule. Use when magnitudes, angle, and direction context are given.

> > **Method 2 — Using the determinant:** > Write the 3×3 determinant with î, ĵ, k̂ in the first row, components of A in the second, components of B in the third. Expand along the first row. > Use when vectors are given in component form.

The geometric meaning of the cross product magnitude

|A × B| = AB sinθ is the area of the parallelogram formed by A and B.

This is not just a curiosity. It is why the cross product appears in area calculations, why angular momentum involves a cross product, why the magnetic force involves a cross product. Whenever physics needs "the area of the region swept out" or "the perpendicular extent of an interaction," a cross product is doing the work.

Comparing the two products

	Dot Product	Cross Product
Symbol	A · B	A × B
Result	Scalar (number)	Vector
Formula	AB cosθ	AB sinθ, perpendicular direction
Maximum when	Parallel (θ = 0°)	Perpendicular (θ = 90°)
Zero when	Perpendicular	Parallel
Commutative?	Yes: A·B = B·A	No: A×B = −(B×A)
Physics use	Work, power, potential	Torque, angular momentum, magnetic force

Common mistakes

Warning: Using sinθ for dot product or cosθ for cross product. The dot product uses cosθ — parallel alignment. The cross product uses sinθ — perpendicular extent. If you mix them, your work calculation will give zero when force and displacement are parallel, which is exactly backwards.

> > **Forgetting the direction of the cross product.** The cross product is a vector. Giving only the magnitude and forgetting the direction is an incomplete answer. Always state the direction using the right-hand rule or the component form result. > > **Getting the right-hand rule wrong.** Curl from the *first* vector to the *second* through the *smaller* angle. If you curl the wrong way, the direction flips. > > **Assuming A × B = B × A.** It does not. The cross product is anticommutative. Order matters — always write the vectors in the specified order.

A worked example

A = 3î + 4ĵ and B = 2î − 3ĵ. Find A · B and A × B.

Dot product: A · B = (3)(2) + (4)(−3) = 6 − 12 = −6

Negative — the vectors lean away from each other. The angle between them is obtuse.

Cross product:

$\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 4 & 0 \\ 2 & -3 & 0 \end{vmatrix}$

= î(4·0 − 0·(−3)) − ĵ(3·0 − 0·2) + k̂(3·(−3) − 4·2) = î(0) − ĵ(0) + k̂(−9 − 8) = −17k̂

The cross product points in the negative z-direction — into the page. Magnitude 17. The right-hand rule confirms: curling from A to B in the xy-plane, the thumb points downward (negative z).

What to remember

The dot product measures how much two vectors agree — how parallel they are. It gives a scalar. Use it for work, power, and finding angles.

The cross product measures how much two vectors disagree — how perpendicular they are. It gives a vector perpendicular to both. Use it for torque, angular momentum, and magnetic force.

The choice between them is not arbitrary. It is dictated by the physics. When the result should be a scalar and depends on alignment — dot. When the result should be a vector and depends on the plane the two vectors define — cross.

The next article introduces calculus into kinematics — specifically what a derivative means physically when applied to position. The dot and cross products will reappear throughout mechanics: work uses the dot product, torque uses the cross product, and the connection between them reveals something deep about energy and rotation.