Physics

1.2 - Combining Vectors

8 min

Resultant by scale drawing

Resolving a forceOutside your course

note 1 of 4

Finding a resultant force by scale drawing

Two or more forces acting on an object can be replaced by a single resultant force that has the same overall effect.
Along the same straight line - Add forces acting in the same direction and subtract forces acting in opposite directions.
At an angle to each other - Use a scale drawing.

Drawing the forces tip-to-tail

Choose a sensible scale (e.g. 1 cm = 1 N).
Draw each force to scale as an arrow, joined tip-to-tail - each arrow starts where the previous one ended.
Draw a straight line from the tail of the first arrow to the tip of the last arrow. This line is the resultant force.
Measure the length of the resultant and use the scale to find its magnitude.
Measure the angle of the resultant with a protractor to find its direction.

The tip-to-tail method works for any number of forces, and for any vector quantity (such as displacement or velocity).

On squared paper, an orange arrow labelled 9 cm points north and a blue arrow labelled 5 cm runs east from its tip, and a green dashed arrow closes the triangle, labelled resultant force 10 cm, with the angle marked 29° bearing from north.

note 2 of 4

Worked example - Resultant force by scale drawing

An object has a force of 5 N acting to the north and a force of 12 N acting to the east. Find the magnitude and direction of the resultant force.

On a grid, an orange arrow points up labelled 5 N, a blue arrow labelled 12 N runs right from its tip, and a green dashed arrow closes the triangle, labelled resultant force.

Step 1 - Choose a scale

Use 1 cm to represent 1 N.

Step 2 - Draw the forces tip-to-tail

Draw the 5 N force as a 5 cm arrow pointing north. From its tip, draw the 12 N force as a 12 cm arrow pointing east.

Step 3 - Draw the resultant

Draw a straight line from the tail of the first arrow to the tip of the last arrow. Measuring gives a length of 13 cm.

Step 4 - Convert back to newtons

Using the scale, 13 cm represents a resultant force of 13 N.

Step 5 - Measure the direction

Measuring the angle with a protractor gives 67 degrees east of north.

The resultant force is 13 N at 67 degrees east of north.

note 3 of 4

Resultant of two perpendicular vectors by calculation

When two vectors act at right angles, their resultant can be calculated rather than drawn to scale.
Drawing them tip-to-tail forms a right-angled triangle with the resultant as the hypotenuse.

Finding the magnitude with Pythagoras' theorem

$a^{2} + b^{2} = c^{2}$

Where:

$a, b$ = The magnitudes of the two perpendicular vectors
$c$ = The magnitude of the resultant

Finding the direction with trigonometry

$\tan θ = \frac{opposite}{adjacent}$

Find the angle using the inverse tangent ( $\tan^{- 1}$ ) on a calculator.
State the direction relative to one of the original vectors or as a bearing.

On a grid, a right-angled triangle with a blue arrow labelled a along the bottom, an orange arrow labelled b up the right side, a green hypotenuse labelled c, and the angle marked theta.

note 4 of 4

Worked example - Resultant by Pythagoras and trigonometry

Two forces act on an object: 5.0 N to the north and 12.0 N to the east. Find the magnitude and direction of the resultant force.

Step 1 - Sketch the triangle

Draw the two forces tip-to-tail and add the resultant from the start of the first to the end of the last. The triangle is right-angled, with the resultant as the hypotenuse. The sketch does not need to be to scale.

Step 2 - Find the magnitude with Pythagoras

With $a = 5.0$ N and $b = 12.0$ N:

5.0² + 12.0² = 25 + 144 = 169

$c = \sqrt{169} = 13.0 N$

Step 3 - Find the direction with trigonometry

The side opposite the angle from north is the 12.0 N force; the adjacent side is the 5.0 N force:

$\tan θ = \frac{12.0}{5.0} = 2.4$

$θ = \tan^{- 1} (2.4) = 67^{\circ} (to 2 s.f.)$

Step 4 - State the answer

The resultant force is 13.0 N at 67 degrees east of north.

key terms

TermDefinition

Resultant forceA single force that has the same effect as all the original forces acting on an object together.

end of notes

Key points to remember

Take a quiz View flashcards

2.1 - Speed, Velocity & DisplacementNext up