A level Maths vectors
Vectors describe size and direction. A level Maths uses them in two and three dimensions for geometry and mechanics.
On this page
- Notation
- Magnitude and direction
- Position vectors
- Geometric problems
- Vectors in mechanics
- Exam tips
- Why vectors are useful
- Worked examples
- Common mistakes
- Practice questions
- Frequently asked questions
- Key takeaways
- Where to go next
- Vectors in geometry proofs
- Position, velocity and acceleration vectors
- How to revise vectors
Notation
Vectors can be written in component form, such as 3i + 4j − k, or as column vectors. i, j and k are unit vectors along the x, y and z axes.
Magnitude and direction
- The magnitude of ai + bj + ck is √(a² + b² + c²).
- A unit vector in the same direction is the vector divided by its magnitude.
- In two dimensions, the direction can be given as an angle using trigonometry.
Position vectors
The position vector of a point A is the vector from the origin to A. The vector from A to B is b − a. The distance AB is the magnitude of that vector.
Geometric problems
- Two vectors are parallel if one is a scalar multiple of the other.
- Points are collinear if the vectors between them are parallel and share a point.
- Midpoints and ratios: the point dividing AB in a given ratio can be found from a and b.
Vectors in mechanics
Forces, velocities and accelerations are vectors. Find a resultant force by adding vectors, and use F = ma in vector form.
Exam tips
- Draw a sketch, even for 3D problems.
- Keep vector notation clear, with bold or underlined letters.
- Check magnitudes are positive and units are included in mechanics questions.
Why vectors are useful
Vectors let you describe both size and direction with a single object. In geometry they make proofs about parallel lines, midpoints and ratios short and clean. In mechanics they represent forces, velocities and accelerations, so adding forces becomes simple addition of components.
Worked examples
Example 1: magnitude
Find the magnitude of 3i + 4j − 12k.
- √(9 + 16 + 144) = √169 = 13.
Example 2: a unit vector
Find a unit vector in the direction of 3i + 4j.
- Magnitude 5, so the unit vector is 0.6i + 0.8j.
Example 3: position vectors
A is (1, 2, 3) and B is (4, 6, 3). Find vector AB, its length and the midpoint of AB.
- AB = b − a = (3, 4, 0), so |AB| = 5.
- Midpoint = ½(a + b) = (2.5, 4, 3).
Example 4: parallel vectors
Show that 2i − 4j + 6k is parallel to i − 2j + 3k.
- 2i − 4j + 6k = 2(i − 2j + 3k), a scalar multiple, so they are parallel.
Example 5: direction in 2D
Find the angle that 3i + 4j makes with i.
- tanθ = 4/3, so θ ≈ 53.1°.
Example 6: resultant force
Forces (2i + 3j) N and (4i − j) N act on a particle. Find the resultant and its magnitude.
- Resultant = 6i + 2j N, with magnitude √40 ≈ 6.32 N.
Common mistakes
- Writing AB = a − b instead of b − a.
- Forgetting the third component in 3D magnitudes.
- Showing vectors are parallel but not that they share a point when proving collinearity.
Practice questions
Try each one before opening the answer.
Q1. Find the distance between (2, −1, 5) and (5, 3, 5).
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Q2. Points A, B, C have position vectors a, a + 2d, a + 5d. Are they collinear?
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Q3. A particle of mass 2 kg has resultant force (6i − 8j) N. Find its acceleration and its magnitude.
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Frequently asked questions
Does A level Maths include the scalar product?
The scalar (dot) product is usually part of Further Maths rather than A level Maths. Check your specification.
How are vectors written in the exam?
Use bold letters in print; in handwriting, underline them. Column vectors are also accepted.
Key takeaways
- A vector has size and direction.
- Magnitude uses Pythagoras in 2D and 3D.
- AB = b − a.
- Parallel vectors are scalar multiples.
- In mechanics, add force vectors to find the resultant.
Where to go next
Vectors support mechanics and geometry.
Vectors in geometry proofs
A classic exam question gives a shape, such as a parallelogram or triangle, labels some vectors, and asks you to prove that points are collinear or that lines are parallel. The method is always the same. Write each required vector as a route through known vectors. If one vector is a multiple of another, they are parallel. If they also share a point, the points lie on one straight line.
Example 7: a midpoint proof
OABC is a parallelogram with OA = a and OC = c. M is the midpoint of AB. Find OM.
- AB = OC = c, so AM = ½c.
- OM = OA + AM = a + ½c.
Example 8: dividing a line in a ratio
P divides AB in the ratio 1 : 3. Write OP in terms of a and b.
- AP = ¼AB = ¼(b − a).
- OP = a + ¼(b − a) = ¾a + ¼b.
Position, velocity and acceleration vectors
In mechanics, velocity and acceleration are written as vectors in i and j. For constant acceleration, the vector form of SUVAT works component by component: r = r0 + ut + ½at². For variable acceleration, differentiate or integrate each component separately. A common question asks when a particle is due north of a point; this means its i component equals that of the point, so set those equal and solve for t.
Example 9: position at a time
A particle starts at (2i + j) m with velocity (3i − 2j) m/s and constant acceleration (i + 2j) m/s². Find its position at t = 2.
- r = (2 + 6 + 2)i + (1 − 4 + 4)j = 10i + j m.
How to revise vectors
Vectors are a short topic, so a few focused sessions go a long way. Practise magnitudes and unit vectors until they are quick, then work through a set of geometry proofs and finish with vector mechanics. Always draw a sketch, even a rough one: it shows you which route between points to take and helps you avoid sign errors.
Once these basics are secure, vectors become one of the most reliable sources of marks on the paper, because the methods are short and predictable.