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URL: https://nakafa.com/en/subjects/physics/vector/multiplication
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/physics/vector/multiplication/en.mdx

Learn the difference between vector dot product and cross product, including their links to work and torque.

---

## 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 |

Visible text: | Type of multiplication | Symbol | Result | Physics example |
| :--------------------- | :----- | :----- | :-------------- |
| dot product | | scalar | work |
| cross product | | 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:

Visible text: The dot product of and is defined as:

```math
\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.

Visible text: Angle 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:

```math
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:

Visible text: A person pulls a box with a force while the box moves . The force makes a angle with the displacement direction. The work is:

```math
\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.

Visible text: If the force is perpendicular to the displacement, and . 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:

Visible text: The cross product of and has magnitude:

```math
\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.

Visible text: The result is a vector perpendicular to the plane containing and . Its direction is determined by the right-hand rule.

Component: Vector3d
Props:
- title: Direction of $$\vec{A}\times\vec{B}$$
  Visible text: Direction of
- description: Vectors $$\vec{A}$$ and $$\vec{B}$$ lie in
one plane. Their cross product points perpendicular to that plane.
  Visible text: Vectors and lie in
one plane. Their cross product points perpendicular to that plane.
- cameraPosition: [8, 7, 9]
- cameraTarget: [1.6, 1.4, 1.2]
- vectors: (() => {
const a = [4, 0, 0];
const b = [0, 3, 0];
const cross = [
a[1] * b[2] - a[2] * b[1],
a[2] * b[0] - a[0] * b[2],
a[0] * b[1] - a[1] * b[0],
];
const displayScale = 4;
const crossDirection = cross.map((component) => component / displayScale);

return [
{
color: getColor("RED"),
from: [0, 0, 0],
label: "A",
labelAnchorX: "center",
labelOffset: [0, -0.45, 0],
labelProgress: 1 / 2,
to: a,
},
{
color: getColor("GREEN"),
from: [0, 0, 0],
label: "B",
labelAnchorX: "center",
labelOffset: [0.5, 0, 0.15],
labelProgress: 1 / 2,
to: b,
},
{
color: getColor("BLUE"),
from: [0, 0, 0],
label: "A x B",
labelAnchorX: "center",
labelOffset: [0.45, 0.35, 0.2],
labelProgress: 1 / 2,
lineWidth: 4,
to: crossDirection,
},
];
})()

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.

Visible text: In this visual, points along the positive axis and points along the positive axis. The cross-product formula gives the positive direction for . 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:

```math
\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.

Visible text: Vector 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 $$\theta=90^\circ$$.

Visible text: A door is pushed with a force at a distance of from the hinge. The force is perpendicular to the door, so .

```math
\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.

Visible text: If the force is applied at a smaller angle, 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 $$\sin0^\circ=0$$.

Visible text: The main difference can be seen from the angle. The dot product is largest when two vectors point the same way because . The cross product is zero in that same condition because .

| 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 |

Visible text: | Angle condition | | |
| :-------------- | :---------------------------------------- | :------------------------------------------------------ |
| | maximum | zero |
| | zero | maximum |
| | negative | zero |

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