Vector-valued position functions | Year 12

01 / LEARN

The idea, step by step

Position depends on the origin

A position function is $r (t) = x (t) i + y (t) j + z (t) k$ . Its components are coordinates measured from a specified origin. A constant z gives a path in a plane parallel to the xy plane.

The magnitude of r is distance from the origin. It is not generally the distance travelled since the start. Displacement is the later position minus the earlier position.

Keep units and parameter meaning

If t is seconds and components are metres, component derivatives have units m/s. A dimensionless parameter derivative is not automatically a physical velocity.

The magnitude of $r (b) - r (a)$ is endpoint separation. Curved travel can have greater length. Returning to the start gives zero displacement even after a long journey.

A common trap

Position and displacement are interchangeable.

Instead: Displacement is a difference of positions and does not depend on moving the origin.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example

For $r = (t^{2}, 2 t, 3)$ m, find position at 2 and displacement from 1 to 2.

Evaluate r (2) and r (1)=(1, 2, 3).
Subtract earlier from later.
The displacement magnitude is $\sqrt{13}$ m.

Position (4, 4, 3) m; displacement (3, 2, 0) m.

Example 2

Worked example

For $(cos t, sin t)$ , describe displacement from 0 to $2 π$ .

Both endpoint positions are (1, 0).
Their difference is zero.
The travelled length is $2 π$ , not zero.

Zero displacement; one unit circle travelled.

Check your understanding

Find displacement from 0 to 2 for $(1 + t, t^{2}, 4 - t)$ .

Show answer

$(2, 4, - 2)$

Subtract (1, 0, 4) from (3, 4, 2).

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Evaluate r (3) for $(2 t, t^{2} - 1, 5)$ .
2 marks
Find displacement from 0 to 1 for $(t, t^{2}, t^{3})$ .
2 marks
Find distance from the origin for $(3 cos t, 3 sin t, 2)$ .
2 marks
Find endpoint separation between (1, 2, 3) and (4, 6, 3).
2 marks
Identify the Cartesian path of $(t^{2}, 2 t)$ .
2 marks
Give a position function for the line through (1, 0, 2) with direction (2, -1, 3).
2 marks

Read the worked solutions

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

Evaluate r (3) for $(2 t, t^{2} - 1, 5)$ .
2 marks
Substitute 3 into each coordinate.
The third coordinate stays 5.
$(6, 8, 5)$
Find displacement from 0 to 1 for $(t, t^{2}, t^{3})$ .
2 marks
The initial vector is zero.
The final vector is (1, 1, 1).
$(1, 1, 1)$
Find distance from the origin for $(3 cos t, 3 sin t, 2)$ .
2 marks
Magnitude squared is $9 ({cos}^{2} t + {sin}^{2} t) + 4$ .
This equals 13.
$\sqrt{13}$
Find endpoint separation between (1, 2, 3) and (4, 6, 3).
2 marks
The displacement is (3, 4, 0).
Its magnitude is 5.
5 units.
Identify the Cartesian path of $(t^{2}, 2 t)$ .
2 marks
$t = y / 2$ .
Substitute into x.
$y^{2} = 4 x$ .
Give a position function for the line through (1, 0, 2) with direction (2, -1, 3).
2 marks
At zero the given point is obtained.
The parameter moves along the required direction.
$r = (1, 0, 2) + t (2, - 1, 3)$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics topic 6.3–6.5: polynomial and trigonometric parameterisations, related coordinate rates, motion, tangent directions and arc length. This is school Vector calculus, not vector-field calculus.

Official curriculum source ↗

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