# Cross Product and Geometry in Space

Mathematics I · Matrices and Vectors · https://tryals.app/learn/mathematics-i/cross-product-and-geometry-in-space

## Multiplying Vectors into a Vector

In 3D space, the **cross product** $u \times v$ multiplies two vectors to produce a **new vector**. Unlike the scalar dot product, it answers geometric questions by yielding a normal direction and a measured area.

| Feature | Cross Product ($u \times v$) | Dot Product ($u \cdot v$) |
| --- | --- | --- |
| **Output** | **Vector** (perpendicular) | **Scalar** |
| **Formula** | $\|u\| \|v\| \sin\theta$ | $\|u\| \|v\| \cos\theta$ |
| **Zero Condition** | Parallel ($u \times v = \mathbf{0}$) | Orthogonal ($u \cdot v = 0$) |

The **magnitude** $\|u \times v\| = \|u\|\|v\|\sin\theta$ equals the **area of the parallelogram** formed by $u$ and $v$. The cross product is largest when vectors are perpendicular, and zero when parallel.

**Common pitfall:** Expecting the cross product to be commutative. It is **anticommutative**: $u \times v = -(v \times u)$. Swapping the order flips the resulting direction.

## Triple Products & Volume

The cross product is strictly limited to 3-dimensional space because it requires a mutually perpendicular third dimension to point into. 

A related quantity, the **scalar triple product** $u \cdot (v \times w)$, yields the **signed volume** of the parallelepiped spanned by three vectors.

| Property | Scalar Triple Product |
| --- | --- |
| **Formula** | $u \cdot (v \times w)$ |
| **Output** | Real number (**volume**) |
| **Coplanar Test** | Equals $0$ when vectors lie in a flat plane |

**Pitfall distinction:** The dot product yields a scalar measuring alignment ($\cos\theta$); the cross product yields a vector measuring area ($\sin\theta$); the scalar triple product combines both to measure 3D volume.

## Practice questions

4 of this lesson's 9 practice questions, with answers. The full set is in the app.

### 1. Sort each pair of vectors by whether its cross product is the zero vector.

**Answer:**

- Cross product is zero: (1, 2, 3) and (2, 4, 6), (1, 1, 1) and (-2, -2, -2), (0, 0, 5) and (0, 0, -3)
- Cross product is nonzero: (1, 0, 0) and (0, 1, 0), (2, 1, 0) and (1, 3, 0)

**Why:** (2, 4, 6) = 2(1, 2, 3), (-2, -2, -2) = -2(1, 1, 1) and (0, 0, -3) is a multiple of (0, 0, 5), so those three pairs are parallel and span no area at all. The other two pairs genuinely span a parallelogram, so their cross products are nonzero.

Page: https://tryals.app/practice/mathematics-i/cross-product-and-geometry-in-space/sort-each-pair-of-vectors-by-whether-its-cross-product-is-the-zero

### 2. Torque is a cross product, and a longer spanner loosens a stubborn bolt more easily. Which feature of the cross product does the spanner exploit?

A. That its direction is perpendicular to both inputs, which aligns the force with the bolt
B. That it is anticommutative, so reversing the grip reverses the torque
C. That it vanishes when the vectors are parallel, reducing wasted effort
D. That its magnitude scales with the distance from the pivot

**Answer:** D. That its magnitude scales with the distance from the pivot

**Why:** You are buying |r|. The magnitude |r||F|sin θ scales with the lever arm, so a handle twice as long delivers twice the torque from the same push. The sin θ term earns its place too: pushing along the spanner rather than across it gives zero, which is why you push perpendicular without thinking about it.

Page: https://tryals.app/practice/mathematics-i/cross-product-and-geometry-in-space/torque-is-a-cross-product-and-a-longer-spanner-loosens-a-stubborn

### 3. Select every TRUE statement about u x v.

A. Its length equals the area of the parallelogram the two vectors span.
B. The cross product is perpendicular to both of its inputs.
C. It vanishes precisely when the two vectors lie along one line.
D. The cross product is commutative: u x v equals v x u.
E. The cross product is defined only in three dimensions.

**Answer:** A. Its length equals the area of the parallelogram the two vectors span.; B. The cross product is perpendicular to both of its inputs.; C. It vanishes precisely when the two vectors lie along one line.

**Why:** The magnitude is the spanned area, it peaks at 90 degrees because sine does, and swapping the order negates it. It returns a vector, not a scalar. And the zero case is PARALLEL, not perpendicular, importing the dot product's vanishing test is the trap this question sets.

Page: https://tryals.app/practice/mathematics-i/cross-product-and-geometry-in-space/select-every-true-statement-about-u-x-v

### 4. The dot product of two vectors is a number; the cross product is a vector. Why does one of them exist in every dimension while the other is peculiar to three?

A. Because only in three dimensions does a plane leave one perpendicular.
B. Because vectors in dimensions above three have no angles between them, so the cross product has nothing to measure.
C. Because the dot product is simpler, and simpler operations generalise while complicated ones do not.
D. Because the cross product requires the right-hand rule, and hands are three-dimensional objects.

**Answer:** A. Because only in three dimensions does a plane leave one perpendicular.

**Why:** The dot product's formula has nothing dimension-specific in it. The cross product is defined by a geometric demand, and in three dimensions a plane leaves exactly one perpendicular line; in four it leaves a whole plane. What generalises is the wedge product, the cross product is a three-dimensional coincidence, and physics uses it because three dimensions is where we live.

Page: https://tryals.app/practice/mathematics-i/cross-product-and-geometry-in-space/the-dot-product-of-two-vectors-is-a-number-the-cross-product-is-a