Cross product and determinant calculations | Year 12

01 / LEARN

The idea, step by step

Construct a perpendicular vector

For $a = (a_{x}, a_{y}, a_{z})$ and $b = (b_{x}, b_{y}, b_{z})$ , the cross product is $(a_{y} b_{z} - a_{z} b_{y}, a_{z} b_{x} - a_{x} b_{z}, a_{x} b_{y} - a_{y} b_{x})$ . It is perpendicular to both inputs. The middle component has the opposite cofactor sign in a determinant expansion.

Order matters: $b \times a = - (a \times b)$ . Use the right-hand rule in a right-handed coordinate system. Parallel vectors give a zero cross product, which has no unit direction.

Check size and orientation

Dot the result with each input; both checks should give zero. The magnitude equals $∣ a ∣ ∣ b ∣ sin θ$ . That geometric expression checks size, but not the orientation sign.

The scalar triple product $a \cdot (b \times c)$ is a determinant. Its absolute value gives a parallelepiped volume. A negative determinant records orientation, not a negative physical volume.

A common trap

Swapping cross-product inputs does not matter.

Instead: It reverses the vector direction; the magnitude alone is unchanged.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Find (1,2,0) cross (0,1,3).

Use the component formula: $(6, - 3, 1)$ .
The dot products with the inputs are $6 - 6 = 0$ and $- 3 + 3 = 0$ .

(6, - 3, 1)

Example 2

Worked example 2

Find $(1, 0, 0) \cdot [(0, 2, 0) \times (0, 0, 3)]$ .

The cross product is $(6, 0, 0)$ .
Its dot product with $(1, 0, 0)$ is 6.

6

Check your understanding

What is (2,4,6) cross (1,2,3)?

Show answer

The zero vector.

The vectors are parallel, so their spanned area is zero.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find (1,0,1) cross (0,2,1).
2 marks
Find (0,2,1) cross (1,0,1).
2 marks
Find $i \times j$ and $j \times i$ .
2 marks
Is (3,0,-1) perpendicular to (1,2,3) cross (2,-1,0)?
2 marks
Find the determinant with rows (1,2,0), (0,1,3), (2,0,1).
2 marks
Find the volume spanned by (2,0,0), (0,3,0), (0,0,4).
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find (1,0,1) cross (0,2,1).
2 marks
Use $(0 \cdot 1 - 1 \cdot 2, 1 \cdot 0 - 1 \cdot 1, 1 \cdot 2 - 0 \cdot 0)$ .
Check both input dot products are zero.
$(- 2, - 1, 2)$
Find (0,2,1) cross (1,0,1).
2 marks
Reverse the order of the previous cross product.
Negate every component.
$(2, 1, - 2)$
Find $i \times j$ and $j \times i$ .
2 marks
The positive basis is right-handed.
Reversing the order negates the result.
$k$ and $- k$
Is (3,0,-1) perpendicular to (1,2,3) cross (2,-1,0)?
2 marks
The cross product is $(3, 6, - 5)$ .
Its dot product with $(3, 0, - 1)$ is 14.
No.
Find the determinant with rows (1,2,0), (0,1,3), (2,0,1).
2 marks
Cross the last two rows: $(1, 6, - 2)$ .
Dot with the first: $1 + 12 = 13$ .
$13$
Find the volume spanned by (2,0,0), (0,3,0), (0,0,4).
2 marks
The determinant is $2 \cdot 3 \cdot 4 = 24$ .
Take its absolute value for volume.
$24$ cubic units

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics vector algebra, vector/Cartesian equations and linear systems. Original explanations and tasks.

Official curriculum source ↗

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