Coaching Centre
Vectors in two dimensions
Adding vectors head to tail and by components, resolving a vector into perpendicular components, bearings, Pythagoras and trigonometry, subtracting vectors, and displacement on a horizontal plane.
Bushwalkers in the Blue Mountains still learn to navigate with a map and compass in case the phone dies. Walk 5 km north, then 12 km east, and you are 13 km from your car on a bearing of 067°. Getting from the first sentence to the second is vector addition.
In a straight line, a sign is enough to show direction. On a flat plane, a vector can point anywhere, so you need two numbers: a size and an angle, or two perpendicular components. This page covers both and how to switch between them.
Describing direction
- Compass directions: N, NE, or angles like N30°E (30° east of north).
- True bearings: three digits, measured clockwise from north. North is 000°, east 090°, south 180°, west 270°. N30°E is 030°; S20°W is 200°.
- Angle from the x-axis: used in maths, measured anticlockwise from east (the positive $x$-direction).
Any of these is fine in an answer as long as it is clear. A bare angle like "67°" is not: 67° from what?
Adding vectors
To add vectors by drawing, place them head to tail: start the second where the first finishes. The resultant runs from the tail of the first to the head of the last. The order doesn't matter: $\vec A + \vec B = \vec B + \vec A$.
When the two vectors are perpendicular, the triangle is right-angled, so use Pythagoras for the size and trigonometry for the angle:
$$R = \sqrt{A^2 + B^2} \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$$A walker goes 5.0 km north, then 12.0 km east. Find their displacement.
Resolving a vector into components
Any vector can be replaced by two perpendicular components that add to give it. For a vector of size $v$:
- angle $\theta$ measured from the $x$-axis: $\;v_x = v\cos\theta$, $\;v_y = v\sin\theta$
- bearing $\theta$ (from north): $\;v_{\text{east}} = v\sin\theta$, $\;v_{\text{north}} = v\cos\theta$
Don't memorise which is sine and which is cosine. The component next to the angle uses cosine; the one opposite it uses sine. Signs come from the direction: a west component is negative if east is positive.
Adding any two vectors by components
When the vectors aren't perpendicular, the head-to-tail triangle isn't right-angled. The reliable method:
- Resolve every vector into east and north components (with signs).
- Add all the east components, then all the north components.
- Recombine with Pythagoras and inverse tan, and state the direction clearly.
A drone flies 10.0 km on a bearing of 030°, then 6.0 km on a bearing of 120°. Find its displacement.
Subtracting vectors
To subtract, add the negative: $\vec A - \vec B = \vec A + (-\vec B)$, where $-\vec B$ is the same size as $\vec B$ but points the opposite way. You'll need this in 1G (change in velocity) and 1H (relative velocity).
- In Resolve, set A to 20.0 at 035°. Check the components match the figure above: east 11.47, north 16.38.
- Set the bearing to 200°. Which components are negative, and why?
- In Add, set A = 5.0 at 000° and B = 12.0 at 090°. The resultant should be 13.00 at 067°.
- Now use the drone example: A = 10.0 at 030°, B = 6.0 at 120°. Check 11.66 at 061°.
- In Subtract, find a pair of vectors where A − B is longer than both A and B.
Test yourself
A boat sails 8.0 km east then 6.0 km south. The size of its displacement is
$\sqrt{8.0^2 + 6.0^2} = 10$ km.
A velocity of 20 m s⁻¹ on a bearing of 060° has an east component of
East $= 20\sin60° = 17.3$ m s⁻¹. The angle is measured from north, so the east side is opposite it.
The bearing of south-west is
Clockwise from north: east 090°, south 180°, south-west 225°.
Two forces of 3 N and 4 N act on an object. Which of these could NOT be the size of their resultant?
The resultant lies between $4 - 3 = 1$ N (opposite) and $4 + 3 = 7$ N (same direction).
Label the vector and its components.
Order the steps for adding two vectors that are not perpendicular.
Vectors are added by placing them head to . The single vector that replaces them is the . A bearing is measured from north. Subtracting a vector is the same as adding its . The component next to the angle uses .
An orienteer runs 1.20 km on a bearing of 040°, then 0.80 km on a bearing of 290°. Calculate the size and direction of their displacement, showing the components.
Leg 1: east $= 1.20\sin40° = 0.771$ km; north $= 1.20\cos40° = 0.919$ km.
Leg 2: east $= 0.80\sin290° = -0.752$ km; north $= 0.80\cos290° = 0.274$ km.
Totals: east $= 0.019$ km; north $= 1.193$ km.
Size $= \sqrt{0.019^2 + 1.193^2} = 1.19$ km. Angle from north $= \tan^{-1}(0.019/1.193) = 0.9°$, so the displacement is 1.19 km on a bearing of about 001°, almost due north.