Calculating the Cross Product with a Determinant

There is no need to memorise a separate formula for the cross product. Set the two vectors under the unit vectors and evaluate it exactly like a 3×3 determinant.

Theorem Setting up the determinant

Row one holds the unit vectors, row two the components of , and row three the components of . The order of the rows matters — swapping and reverses the answer.

Concept Cover a column to get each part

To find the

part, cover the

column and the top row. The four numbers left form a 2×2 determinant. Repeat for

and for

The signs alternate: +i, then −j, then +k. Forgetting the minus in front of is the single most common error here.

Example U = (3, −2, 1) and V = (3, 3, 1)

The i part — cover column i:

The j part — cover column j, and take the negative:

The k part — cover column k:

⟹ ⟨−5, 0, 15⟩

A zero component is perfectly normal — it simply means the result has no extent in that direction.

Example U = (4, 2, −1) and V = (5, 1, 4)

The i part:

The j part:

The k part:

⟹ ⟨9, −21, −6⟩

Notice how the negative components feed through: becomes , and the minus outside the bracket flips the whole result.

Note The procedure in five steps

1. Build the matrix: row one

; row two

; row three

2. Cover the i column and take the 2×2 determinant — this is the i part, positive.

3. Cover the j column, take the determinant, then negate it.

4. Cover the k column and take the determinant — positive again.

5. Each 2×2 determinant is the main diagonal minus the other diagonal.

Summary

A cross product is evaluated like a 3×3 determinant.
Row one is i, j, k; row two is U; row three is V.
Cover a column to get that component as a 2×2 determinant.
The signs run +i, −j, +k — never omit the minus on j.
(3, −2, 1) × (3, 3, 1) = ⟨−5, 0, 15⟩.