Vector operations | Year 12

01 / LEARN

The idea, step by step

Operate component by component

Add and subtract matching components; multiply every component by a scalar. The rules describe independent displacements along each axis. A vector does not change when it is translated parallel to itself.

The midpoint of A and B has position $(a + b) / 2$ . More generally, $a + t (b - a)$ is a fraction t of the way from A to B when $0 \leq t \leq 1$ .

Connect algebra and geometry

Nonzero vectors are parallel exactly when one is a scalar multiple of the other. A negative multiple points in the opposite direction. For collinearity of points, compare displacement vectors from the same starting point.

If $A P : P B = m : n$ internally, use $p = (n a + m b) / (m + n)$ . Positive weights place P between A and B. Check the direction and the relative lengths before accepting a ratio answer.

A common trap

The ratio weights multiply the matching endpoint.

Instead: Internal division uses opposite weights: the point nearer A gives A the larger weight.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

For $a = (1, 2, - 1)$ and $b = (3, - 1, 4)$ , find $2 a - b$ .

Double a to get $(2, 4, - 2)$ .
Subtract the corresponding components of b.

(- 1, 5, - 6)

Example 2

Worked example 2

A=(1,0,2), B=(7,3,-1). Find P with AP:PB=1:2 internally.

The displacement AB is $(6, 3, - 3)$ .
Travel one third, then add $(2, 1, - 1)$ to A.

(3, 1, 1)

Check your understanding

Are $(2, - 4, 6)$ and $(- 1, 2, - 3)$ parallel?

Show answer

Yes.

The first is -2 times the second.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

For a=(2,-1,3), b=(-1,4,2), find a+b and 3a-2b.
2 marks
Find the midpoint of (2,4,-2) and (6,-2,8).
2 marks
Find P one quarter of the way from (0,1,2) to (8,5,-2).
2 marks
Are A(1,1,0), B(3,5,2), C(4,7,3) collinear?
2 marks
Solve $2 p + (1, - 1, 3) = (7, 5, - 1)$ .
2 marks
For parallelogram ABCD in order, A(1,0,1), B(3,2,0), C(6,1,4), find D.
2 marks

Read the worked solutions

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

For a=(2,-1,3), b=(-1,4,2), find a+b and 3a-2b.
2 marks
Add components.
For the second result subtract $(- 2, 8, 4)$ from $(6, - 3, 9)$ .
$(1, 3, 5)$ and $(8, - 11, 5)$
Find the midpoint of (2,4,-2) and (6,-2,8).
2 marks
Add coordinates to get $(8, 2, 6)$ .
Divide by 2.
$(4, 1, 3)$
Find P one quarter of the way from (0,1,2) to (8,5,-2).
2 marks
One quarter of displacement $(8, 4, - 4)$ is $(2, 1, - 1)$ .
Add this to the first point.
$(2, 2, 1)$
Are A(1,1,0), B(3,5,2), C(4,7,3) collinear?
2 marks
AB= $(2, 4, 2) = 2 (1, 2, 1)$ .
AC= $3 (1, 2, 1)$ , so the displacements are parallel.
Yes.
Solve $2 p + (1, - 1, 3) = (7, 5, - 1)$ .
2 marks
Subtract the fixed vector to obtain $2 p = (6, 6, - 4)$ .
Divide by 2.
$p = (3, 3, - 2)$
For parallelogram ABCD in order, A(1,0,1), B(3,2,0), C(6,1,4), find D.
2 marks
AD=BC, so $d = a + c - b$ .
Substitute and combine components.
$(4, - 1, 5)$

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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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