Magnitude and direction | Year 11

01 / LEARN

The idea, step by step

A directed change

A vector records a directed change, such as moving 3 metres east and 4 metres north. The arrow's length represents its magnitude, while its arrowhead indicates direction. Magnitude is a non-negative scalar: two journeys can have the same length but different displacements.

For a Cartesian vector $(a, b)$ , the components are horizontal and vertical changes. Apply Pythagoras to its right-angled component triangle. A direction angle is measured anticlockwise from the positive x-axis unless another convention is stated.

∣ (a, b) ∣ = \sqrt{a^{2} + b^{2}}

Choose the quadrant before using a calculator

A calculator's inverse tangent may return a reference angle rather than the required direction. First sketch the signs of the components, then place the direction in its correct quadrant. A direction can also be stated as an angle north of west, for example.

The zero vector has magnitude zero and no unique direction. Dividing a non-zero vector by its magnitude preserves its direction and produces a unit vector. Keep exact roots until a decimal approximation is actually required.

A common trap

Using a negative component as a negative vector length, or accepting the raw inverse-tangent angle.

Instead: Magnitude uses squared components. Determine the quadrant from the component signs before choosing a direction angle.

02 / WORKED EXAMPLES

See the thinking

Example 1

Magnitude and a first-quadrant direction

Find the magnitude and direction of $(3, 4)$ ; give the direction to ${0.1}^{\circ}$ .

Use Pythagoras: $\sqrt{3^{2} + 4^{2}} = 5$ .
Both components are positive, so the direction is in quadrant I.
$θ = {tan}^{- 1} (4 / 3) = 53.130^{\dots \circ}$ .

Magnitude

5

; direction

{53.1}^{\circ}

Example 2

A north-west displacement

A walker has displacement $(- 6, 6)$ km. Give its exact magnitude and direction from the positive x-axis.

Magnitude is $\sqrt{36 + 36} = 6 \sqrt{2}$ km.
Equal absolute components give a $45^{\circ}$ reference angle.
The vector points into quadrant II, so the required direction is $180^{\circ} - 45^{\circ}$ .

6 \sqrt{2}

km at

135^{\circ}

Check your understanding

What are the magnitude and direction of $(0, - 7)$ ?

Show answer

Magnitude $7$ ; direction $270^{\circ}$ .

The arrow points straight down the negative y-axis; its length is seven units.

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the magnitude of $(5, 12)$ .
2 marks
Find the exact magnitude and direction of $(- 4, 4)$ .
2 marks
A displacement is $(8, - 6)$ m. Find its magnitude and its direction south of east to ${0.1}^{\circ}$ .
3 marks
A non-zero vector has magnitude $10$ and direction $60^{\circ}$ . Find its exact components.
3 marks
Find $k$ if $(k, 4)$ has magnitude $5$ .
3 marks
Explain why $(0, 0)$ cannot be assigned a unique direction angle.
2 marks

Read the worked solutions

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

Find the magnitude of $(5, 12)$ .
2 marks
$\sqrt{25 + 144} = 13$ .
$13$
Find the exact magnitude and direction of $(- 4, 4)$ .
2 marks
The magnitude is $\sqrt{32}$ .
A quadrant-II arrow with equal component sizes makes angle $135^{\circ}$ .
$4 \sqrt{2}$ and $135^{\circ}$ .
A displacement is $(8, - 6)$ m. Find its magnitude and its direction south of east to ${0.1}^{\circ}$ .
3 marks
$\sqrt{64 + 36} = 10$ .
${tan}^{- 1} (6 / 8) = 36.869^{\dots \circ}$ .
$10$ m; ${36.9}^{\circ}$ south of east.
A non-zero vector has magnitude $10$ and direction $60^{\circ}$ . Find its exact components.
3 marks
Horizontal component: $10 cos 60^{\circ} = 5$ .
Vertical component: $10 sin 60^{\circ} = 5 \sqrt{3}$ .
$(5, 5 \sqrt{3})$ .
Find $k$ if $(k, 4)$ has magnitude $5$ .
3 marks
$k^{2} + 16 = 25$ , so $k^{2} = 9$ .
Both horizontal directions have the required magnitude.
$k = 3$ or $k = - 3$ .
Explain why $(0, 0)$ cannot be assigned a unique direction angle.
2 marks
Every angle would produce the same zero displacement when the magnitude is zero.
It represents no directed movement, so no direction is distinguished.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: vectors in the plane. Covers magnitude, components, geometry, scalar products, projections and proofs; follows the approved Specialist preparation structure.

Official curriculum source ↗

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