Geometric areas and perpendicular vectors | Year 12

01 / LEARN

The idea, step by step

Area from sides

The area of a parallelogram with sides a and b is $∣ a \times b ∣$ . A triangle using the same two sides has half this area. Use displacements from a common vertex, not unrelated position vectors.

For triangle ABC, take AB and AC. Swapping them reverses the normal but leaves its magnitude unchanged. A zero cross product means collinear vertices and zero triangle area.

Choose the requested normal

Any nonzero multiple of a normal is also normal. For a unit normal, divide by its magnitude. Unless an orientation is specified, both opposite unit normals are valid.

If a normal of specified length k is requested, multiply a unit normal by k. Check its dot products with both side directions and its length. Areas use square units; normals themselves represent a direction rather than an area.

A common trap

The cross-product magnitude is a triangle area.

Instead: It is a parallelogram area; halve it for a triangle.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Find the area of triangle A(0,0,0), B(2,0,0), C(0,3,1).

AB cross AC is $(0, - 2, 6)$ , with magnitude $2 \sqrt{10}$ .
Halve the parallelogram area.

\sqrt{10}

square units

Example 2

Worked example 2

Find both unit normals to (1,0,1) and (0,1,1).

Their cross product is $(- 1, - 1, 1)$ .
Divide by its magnitude $\sqrt{3}$ and allow either sign.

\pm (- 1, - 1, 1) / \sqrt{3}

Check your understanding

Do O, (1,2,3), (2,4,6) form a non-degenerate triangle?

Show answer

No.

Their side vectors are parallel.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Find the parallelogram area spanned by (1,0,0) and (0,2,2).
2 marks
Find triangle area for A(1,0,0), B(3,0,0), C(1,4,0).
2 marks
Give a length-5 vector perpendicular to (1,0,0) and (0,1,0).
2 marks
Find triangle area for sides (1,2,0) and (0,1,3).
2 marks
Explain why replacing b by 3b triples the spanned parallelogram area.
2 marks
Find a unit normal to sides (2,0,0), (0,3,4) with positive z component.
2 marks

Read the worked solutions

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

Find the parallelogram area spanned by (1,0,0) and (0,2,2).
2 marks
The cross product is $(0, - 2, 2)$ .
Its magnitude is $\sqrt{8}$ .
$2 \sqrt{2}$ square units
Find triangle area for A(1,0,0), B(3,0,0), C(1,4,0).
2 marks
The perpendicular sides have lengths 2 and 4.
Use half their product.
$4$ square units
Give a length-5 vector perpendicular to (1,0,0) and (0,1,0).
2 marks
The cross-product direction is the z axis.
Scale its unit direction by 5.
$(0, 0, 5)$ or $(0, 0, - 5)$
Find triangle area for sides (1,2,0) and (0,1,3).
2 marks
Their cross product is $(6, - 3, 1)$ .
Its squared magnitude is 46; halve the magnitude.
$\sqrt{46} / 2$ square units
Explain why replacing b by 3b triples the spanned parallelogram area.
2 marks
Use linearity of the cross product.
Use $∣ 3 v ∣ = 3 ∣ v ∣$ .
The cross product becomes $3 (a \times b)$ , so its magnitude triples.
Find a unit normal to sides (2,0,0), (0,3,4) with positive z component.
2 marks
The cross product is $(0, - 8, 6)$ with length 10.
Normalise using the positive-z orientation.
$(0, - 4 / 5, 3 / 5)$

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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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