Cross Product of Three Vectors- Vector Calculus

What Is the Cross Product of Three Vectors?

The cross product of three vectors is officially called the scalar triple product (or mixed triple product). You calculate it as:

(a × b) · c

It gives you a scalar value — a number, not another vector. That number represents the volume of the parallelepiped formed by the three vectors. Simple as that.

People sometimes confuse this with the vector triple product, which is a × (b × c) and produces a vector. Different thing entirely. Keep them separate.

The Math Behind the Scalar Triple Product

If your vectors are:

a = (a₁, a₂, a₃)
b = (b₁, b₂, b₃)
c = (c₁, c₂, c₃)

The scalar triple product is the determinant:

| a₁ a₂ a₃ |
| b₁ b₂ b₃ |
| c₁ c₂ c₃ |

You expand this along the first row:

a₁(b₂c₃ - b₃c₂) - a₂(b₁c₃ - b₃c₁) + a₃(b₁c₂ - b₂c₁)

That's the formula. Memorize it or keep this page open.

Key Properties

• Cyclic permutation is fine: (a × b) · c = (b × c) · a = (c × a) · b
• Sign flips if you swap vectors: (a × b) · c = - (b × a) · c
• Zero if any two vectors are parallel — the volume collapses to nothing
• Zero if the three vectors are coplanar — they lie flat, no volume
• Absolute value gives volume: the scalar itself can be negative depending on orientation, but |(a × b) · c| is always the actual volume

Scalar Triple Product vs. Vector Triple Product

Stop mixing these up. Here's the difference:

This article focuses on the scalar triple product. That's what 99% of people mean when they ask about "cross product of three vectors."

How to Calculate It — Step by Step

Let's work through an example:

a = (1, 2, 3)
b = (4, 5, 6)
c = (7, 8, 9)

Method 1: Determinant

Set up the matrix:

| 1 2 3 |
| 4 5 6 |
| 7 8 9 |

Calculate the determinant:

= 1(5×9 - 6×8) - 2(4×9 - 6×7) + 3(4×8 - 5×7)
= 1(45 - 48) - 2(36 - 42) + 3(32 - 35)
= 1(-3) - 2(-6) + 3(-3)
= -3 + 12 - 9
= 0

The result is zero. This means the three vectors are coplanar — they all lie in the same plane.

Method 2: Component Formula

Plug directly into:

a₁(b₂c₃ - b₃c₂) - a₂(b₁c₃ - b₃c₁) + a₃(b₁c₂ - b₂c₁)

= 1(5×9 - 6×8) - 2(4×9 - 6×7) + 3(4×8 - 5×7)
= -3 + 12 - 9
= 0 ✓

Same answer. Use whichever method feels less error-prone for you.

Getting Started: Quick Reference

• Write your three vectors in component form: (x, y, z)
• Set up the 3×3 determinant with each vector as a row (or column — it works either way)
• Calculate the determinant using cofactor expansion or row reduction
• Interpret the result: positive = right-hand orientation, negative = left-hand orientation, zero = coplanar vectors

Where You'll Actually Use This

The scalar triple product shows up in:

• Volume calculations in geometry and physics — if you need the volume of a parallelepiped, this is your tool
• Checking coplanarity — if the triple product equals zero, the vectors lie in the same plane
• Determinant-based proofs in linear algebra
• Engineering mechanics — moment calculations involving three vectors

Common Mistakes to Avoid

• Forgetting the dot product at the end — (a × b) gives a vector, you still need to dot that with c
• Confusing scalar and vector triple products — they are not interchangeable
• Sign errors in the determinant — the alternating signs in expansion trips people up constantly
• Using the wrong determinant expansion — stick to the first row unless you're comfortable with cofactors

The Bottom Line

The cross product of three vectors — the scalar triple product — is a determinant that gives you volume. That's it. The math is straightforward once you stop overcomplicating it.

Master the determinant calculation. Know that cyclic permutations don't change the result. Remember that zero means coplanar. Those three things cover 90% of what you'll ever need.