Geometry & Measures
Grade 7-9Vectors
Vectors describe a movement with both a size and a direction, and GCSE questions test them in two very different ways: calculating with column vectors, and using vectors written in terms of a and b to prove facts about lines and shapes. This lesson covers vector arithmetic, magnitude, and the geometric proof style questions that carry the most marks.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Column vectors and vector arithmetic
A column vector shows the horizontal and vertical movement between two points, written as a top number for across and a bottom number for up, with negative numbers for left or down movements. Vectors add and subtract component by component, and multiplying a vector by a scalar (an ordinary number) multiplies each component by that scalar.
If A is at (1, 2) and B is at (6, 5), the column vector AB is found by subtracting the coordinates of A from the coordinates of B, giving 5 across and 3 up. If a = (3, -2) and b = (-1, 4), then a + b = (2, 2), found by adding the top numbers and the bottom numbers separately.
Magnitude of a vector
The magnitude of a vector, written using vertical bars around the letter, is its length, found using Pythagoras theorem on the horizontal and vertical components. Square each component, add the results, then take the square root.
For a = (6, 8), the magnitude is the square root of 6 squared plus 8 squared, which is the square root of 100, giving 10. When the components do not produce a whole number, the answer is left as a simplified surd rather than rounded.
Vectors in terms of a and b
In geometry proof questions, sides of a triangle or other shape are given as letters such as a and b rather than numbers, and every other vector in the diagram must be built up by travelling along known routes. Moving backwards along a labelled arrow reverses its sign.
In triangle OAB with OA = a and OB = b, the vector AB is found by travelling from A back to O, then on to B, which gives AB = negative a plus b, more commonly written as b minus a.
Ratios, midpoints and parallel vectors
When a point divides a line in a given ratio, only that fraction of the vector along that line is travelled. A midpoint always corresponds to a fraction of one half. Two vectors are parallel exactly when one is a scalar multiple of the other, and this is the key fact used to prove that two lines in a diagram are parallel or that three points lie on a straight line.
If M is the midpoint of AB, then AM is half of AB, so OM = OA + half of AB = a + half of (b minus a), which simplifies to half of a plus b. If a second vector works out to be exactly a number multiplied by this same vector, the two lines are parallel, and that number gives the ratio between their lengths.
Worked Examples
Three exam-style questions, fully solved.
a = (6, 8). Find the magnitude of a, |a|.
Easy- 1.Square each component and add: 6² + 8²
- 2.Take the square root: √100
Answer: |a| = 10
In triangle OAB, OA = a and OB = b. P is the point on AB such that AP : PB = 2 : 1. Find the vector OP in terms of a and b.
Medium- 1.Travel from O to A, then along the fraction 2/3 of AB: OP = OA + AP = a + (2/3)(b - a)
- 2.Expand and simplify: OP = a + (2/3)b - (2/3)a
Answer: OP = (1/3)a + (2/3)b
OA = a and OB = b. C is the point on OA with OC : CA = 3 : 2. D is the point on OB with OD : DB = 3 : 2. Show that CD is parallel to AB, and find the ratio CD : AB.
Hard- 1.Write OC and OD as fractions of a and b: OC = (3/5)a, OD = (3/5)b
- 2.Find CD: CD = OD - OC = (3/5)b - (3/5)a = (3/5)(b - a)
- 3.Find AB for comparison: AB = OB - OA = b - a, so CD = (3/5) × AB
Answer: CD is parallel to AB, with CD : AB = 3 : 5
Avoid These
The most common mistakes students make.
Subtracting the coordinates the wrong way round when finding a column vector, rather than always taking the end point minus the start point.
Using the wrong fraction for a ratio split, such as using one half for a point that divides a line 2 to 1 rather than the correct fraction of two thirds.
Leaving an answer only partly simplified, such as stopping at a + (2/3)b - (2/3)a instead of collecting the a terms together.
Stating that two lines are parallel without showing the scalar multiple relationship between their vectors as evidence.
Forgetting to take the square root at the end when finding a magnitude, and leaving the answer as the sum of the squared components.
FAQ
Questions parents and students ask.
Before this topic, make sure you know
What to learn next
Want a plan built around your child specifically?
Get our free 8-video course, or book a free Roadmap Call for a personalised plan.