# Dot Product, Cross Product, and Applications

Physics I · Linear Algebra and Geometry · https://tryals.app/learn/physics-i/dot-product-cross-product-and-applications

## Dot Product & Cross Product

The dot product measures **alignment**; the cross product measures **perpendicularity**. Picking the wrong one silently zeroes out your geometry.

**Dot Product** (scalar result):
$$\vec{u}\cdot\vec{v} = u_xv_x + u_yv_y + u_zv_z$$
$$\vec{u}\cdot\vec{v} = \|\vec{u}\|\|\vec{v}\|\cos\theta$$

**Cross Product** (vector result, 3D only):
$$\vec{u}\times\vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_x & u_y & u_z \\ v_x & v_y & v_z \end{vmatrix}$$

**Common pitfall:** The dot product peaks for **parallel** vectors and dies at 90 degrees; the cross product does the exact opposite.

## Properties & Applications

| Feature | Dot Product | Cross Product |
| --- | --- | --- |
| **Result** | Scalar | Vector (3D) |
| **Symmetry** | Commutative | Anti-commutative |
| **Geometry** | Projection | Area |
| **Zero at** | Perpendicular ($\perp$) | Parallel ($\parallel$) |

**Physics Applications:**
- **Work:** $W = \vec{F}\cdot\vec{d}$
- **Torque:** $\vec{\tau} = \vec{r}\times\vec{F}$
- **Magnetic Force:** $\vec{F} = q\vec{v}\times\vec{B}$
- **Angular Momentum:** $\vec{L} = \vec{r}\times\vec{p}$

## Practice questions

4 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. Sort these properties into Dot Product or Cross Product.

**Answer:**

- Dot Product: Result is a scalar, Commutative: u·$v = v\cdot u$, Measures angle between vectors
- Cross Product: Magnitude is $|u||v|\sin\theta$, Anti-commutative: u×$v = -(v\times u)$

**Why:** Dot product gives a scalar measuring alignment; cross product gives a perpendicular vector.

Page: https://tryals.app/practice/physics-i/dot-product-cross-product-and-applications/sort-these-properties-into-dot-product-or-cross-product

### 2. You pull a sled with force 100 N at some angle to the ground, moving it 10 m. Which single quantity decides how much **work** ($\vec{F}\cdot\vec{d}$) you did?

A. The angle of your pull
B. The time the pull lasted
C. The total force, direction is irrelevant to work
D. The sled’s weight

**Answer:** A. The angle of your pull

**Why:** The dot product is a **projection**: it pays only for the force component along the motion. At $0°$ you get all 1000 J; at $90°$, nothing. Whenever physics asks "how much of this acts along that?", a dot product is the answer.

Page: https://tryals.app/practice/physics-i/dot-product-cross-product-and-applications/you-pull-a-sled-with-force-100-n-at-some-angle-to-the-ground-moving

### 3. The dot product vanishes for perpendicular vectors and the cross product vanishes for parallel ones. Why are the two conditions opposite?

A. Because the dot product is a projection and the cross product is its inverse
B. Because the cross product is only defined for non-parallel vectors
C. Because one uses cosine and the other sine, which is a notational accident
D. Because the dot measures the shared component and the cross the area

**Answer:** D. Because the dot measures the shared component and the cross the area

**Why:** Alignment and spread are complementary. That is why the two vanishing conditions are opposite, and why $\cos$ and $\sin$ appear where they do.

Page: https://tryals.app/practice/physics-i/dot-product-cross-product-and-applications/the-dot-product-vanishes-for-perpendicular-vectors-and-the-cross

### 4. Order the steps to find the angle between u = (1,0) and v = (1,1).

**Answer:**

1. Compute u · $v = 1\times1 + 0\times1 = 1$
2. Compute ‖u‖ = 1 and ‖v‖ = $\sqrt2$
3. cos $\theta$ = (u$\cdot v)$/(‖u‖‖v‖) = 1/$\sqrt2$
4. $\theta = 45°$ ($\pi/4$ radians)

**Why:** The dot product formula directly gives the cosine of the angle.

Page: https://tryals.app/practice/physics-i/dot-product-cross-product-and-applications/order-the-steps-to-find-the-angle-between-u-1-0-and-v-1-1