# Parallelepiped: Definition, Volume & Scalar Triple Product

## What Is a Parallelepiped?

A **parallelepiped** is a solid figure whose six faces are all **parallelograms**, arranged so that opposite faces are parallel and identical. It is the 3D analogue of a parallelogram: just as a parallelogram is a "slanted rectangle," a parallelepiped is a "slanted box." A cube, a rectangular box ( **cuboid**), and a **rhombohedron** are all special cases where the faces become squares, rectangles, or rhombi.

A parallelepiped has **8 vertices, 12 edges, and 6 faces**. The three edges meeting at any one vertex fully determine the solid — give those three edges as vectors and the whole shape is fixed.

## The Volume Formula: Scalar Triple Product

The **volume** is the absolute value of their **scalar triple product**:

V=∣a⃗⋅(b⃗×c⃗)∣

The **cross product** b⃗×c⃗ produces a vector perpendicular to the base parallelogram, and its length equals the base area. Taking the **dot product** with a⃗ multiplies that base area by the height of a⃗ above the base. Base area times height is volume — exactly what the formula computes.

The absolute value matters: the triple product can come out negative, but volume is never negative, so we take the magnitude. The scalar triple product also equals a **determinant**:

V=∣det⁡\[\begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix}\]∣

## Examples of a Parallelepiped

### Example 1

**Find the volume of a parallelepiped with edge vectors a⃗=(1,0,0), b⃗=(0,2,0), c⃗=(0,0,3).**

These edges are mutually perpendicular, so this is a rectangular box. Using the determinant:

V=∣det⁡\[\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}\]∣

V=∣1×(2×3−0)∣=∣6∣=6

The volume is **6 cubic units**.

### Example 2

**Compute the volume for a⃗=(2,1,0), b⃗=(1,3,0), c⃗=(0,0,4).**

First expand the determinant along the third row:

V=∣det⁡\[\begin{bmatrix} 2 & 1 & 0 \\ 1 & 3 & 0 \\ 0 & 0 & 4 \end{bmatrix}\]∣

V=∣4×(2×3−1×1)∣=∣20∣=20

The volume is **20 cubic units**.

### Example 3: The order-of-operations trap

**A student computes V=∣a⃗⋅(b⃗×c⃗)∣ but does the dot product first.**

The correct way is to respect the bracket: the **cross product must come first**.

### Example 4

**Find the volume for a⃗=(1,2,3), b⃗=(2,0,1), c⃗=(0,1,2).**

V=∣det⁡\[\begin{bmatrix} 1 & 2 & 3 \\ 2 & 0 & 1 \\ 0 & 1 & 2 \end{bmatrix}\]∣

V=∣−3∣=3

The volume is **3 cubic units**.

### Example 5

**Show that if the three edge vectors are coplanar, the parallelepiped has zero volume.**

V=∣det⁡\[\begin{bmatrix} 1 & 1 & 0 \\ 2 & 0 & 0 \\ 0 & 3 & 0 \end{bmatrix}\]∣=0

### Example 6

**A crystallographer models a mineral cell as a parallelepiped with edges a⃗=(3,0,0), b⃗=(1,4,0), c⃗=(0,0,5). Find its volume.**

V=∣5×(3×4−0×1)∣=60

The unit-cell volume is **60 cubic ångströms**.

## Where the Parallelepiped Earns Its Keep

- **Crystallography:** The **unit cell** of a crystal is almost always a parallelepiped.
- **Computer graphics and physics:** A "bounding box" that must hug a rotated object is an oriented parallelepiped.
- **Linear algebra:** The triple product's link to the determinant means the volume of a parallelepiped is the geometric meaning of a 3×3 determinant.

## The Mistakes Students Make Most Often

### Mistake 1: Doing the dot product before the cross product

**Where it slips in:** Computing the dot first.

### Mistake 2: Forgetting the absolute value

**Where it slips in:** The triple product returns a negative number.

### Mistake 3: Confusing a parallelepiped with a rectangular box

**Where it slips in:** Assuming edges are perpendicular.

## Conclusion

- A **parallelepiped** is a 3D solid.
- Its **volume** is the absolute value of the scalar triple product of the three edge vectors.
- The most common mistakes are doing the dot before the cross and dropping the absolute value.

## A Practical Next Step

Work through the exercises below to solidify your understanding:

1. Find the volume of the parallelepiped with edges (2,0,0), (0,3,0), (0,0,1). _(Answer: 6 cubic units.)_
2. Compute ∣a⃗⋅(b⃗×c⃗)∣ for a⃗=(1,0,1), b⃗=(0,1,0), c⃗=(1,0,0). _(Answer: 1 cubic unit.)_
3. What does a determinant of 0 tell you about the three edge vectors? _(Answer: they are coplanar, so the volume is zero.)_