Vectors: translations and vector arithmetic
Describing a translation as a column vector, and adding, subtracting and scaling vectors
About three lessons of 45 to 60 minutes
How do you give a set of directions that works for every point on a shape at once?
"Walk 3 steps right and 2 steps back" is a single instruction, but it moves EVERY point of a shape the same way, all at once. That single instruction is a vector: one pair of numbers that captures a whole movement, direction and distance together, written compactly as the column vector (dx, dy)ᵀ rather than two separate sentences.
Vectors are not just for translating shapes. A satnav's route, a video game character's movement, and a plane correcting for wind are all vector problems: you can add two vectors together (walk one instruction, then another) or scale a vector up or down (walk the same direction twice as far). This unit builds that single idea, description, addition, subtraction and scaling, entirely with whole-number coordinates.
- A ship steering 4 km east and 3 km northwritten compactly as the column vector (4, 3)ᵀ, a single object combining direction and distance
- A game character translated across the screenevery part of the sprite moves by the same vector, exactly like a shape's vertices
- Walking one vector, then anothervector ADDITION: the two instructions combine into a single overall displacement
- Doubling a delivery drone's flight pathSCALAR multiplication: the same direction, twice the distance
What students will be able to do
Students will describe the translation between two points as a column vector, apply a given vector to translate a point or a whole shape, and add, subtract and scale column vectors component by component.
- I can find the column vector (dx, dy)ᵀ that translates one given point to another, by subtracting coordinates.
- I can apply a given column vector to a point, or every vertex of a shape, to find the translated image.
- I can add or subtract two column vectors by adding or subtracting their matching components.
- I can multiply a column vector by a scalar by multiplying every component by that scalar.
- I can find a linear combination of two vectors, such as 2a - b, by scaling first and then adding.
Standards this unit teaches
- GCSE Geometry & Measures, Vectors #24UK GCSE Mathematics (DfE, England)Describing translations as 2D vectors
Subject content statement (Department for Education, "GCSE mathematics: subject content and assessment objectives", published 1 November 2013, reference DFE-00233-2013, "Geometry and measures" section, "Vectors" sub-heading, item 24, https://www.gov.uk/government/publications/gcse-mathematics-subject-content-and-assessment-objectives): students should "describe translations as 2D vectors". This is standard-type content, so ALL GCSE students (Foundation and Higher tier) are taught and assessed on it.
- GCSE Geometry & Measures, Vectors #25UK GCSE Mathematics (DfE, England)Vector addition, subtraction, scalar multiplication and representations
Subject content statement (same document, item 25): students should "apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors". This is underlined-type content, meaning all students are ASSESSED on it (Foundation and Higher tier alike). Item 25 continues, in bold type (Higher tier only, and deliberately NOT covered by this unit or its worksheets), "use vectors to construct geometric arguments and proofs".
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Coordinates in the glossarya vector is just a pair of numbers, exactly like a coordinate, but describing a MOVE rather than a position
- Translation in the glossarythe shape-moving transformation this unit gives a formal vector notation to
- Grade 7 transformations worksheettranslating, reflecting and rotating points on the coordinate plane, one grade earlier and without vector notation or arithmetic
Words to teach and display
- Vector
- a quantity with both a size and a direction, written here in compact column notation as (dx, dy)ᵀ to describe a movement
- Column vector
- a vertical vector; in compact one-line notation, (3, -2)ᵀ means the column whose top entry is 3 and bottom entry is -2
- Translation
- a transformation that slides every point of a shape by the same vector, without turning or resizing it
- Image
- the new point or shape produced after a transformation, often labelled with a dash, e.g. A maps to A'
- Scalar
- an ordinary number (not a vector) used to stretch or shrink a vector by multiplying every component by it
- Resultant vector
- the single vector produced by adding two or more vectors together
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Describing a translation as a column vector
ConcreteGiven a starting point and its image after a translation, the column vector is found by subtracting: dx = (new x) - (old x), and dy = (new y) - (old y). The order matters: always image minus original, never the other way round.
For example, if point A(2, -1) is translated to A'(5, 3), then dx = 5 - 2 = 3 and dy = 3 - (-1) = 4, so the translation vector is (3, 4)ᵀ. Checking it the other way confirms this: starting at (2, -1) and moving 3 right, 4 up lands exactly on (5, 3).
Point B is at (-3, 4). It is translated to the image point B' at (1, 0). Write the translation as a column vector.
- dx = 1 - (-3) = 4.
- dy = 0 - 4 = -4.
- The translation vector is (4, -4)ᵀ.
Answer: (4, -4)ᵀ
- Why is it (new) - (old), and not (old) - (new)?
- If a translation vector is (0, 5)ᵀ, what happens to the x-coordinate of every point?
2. Applying a vector to translate a point or shape
PictorialGoing the other way, given a point and a vector, ADD the vector's components to the point's coordinates to find the image. Applied to a whole shape, the SAME vector is added to every vertex, which is exactly why the shape does not change size or turn, only position.
For triangle A(1, 1), B(3, 1), C(2, 3) translated by the vector (2, -3)ᵀ: A' = (1 + 2, 1 - 3) = (3, -2). B' = (3 + 2, 1 - 3) = (5, -2). C' = (2 + 2, 3 - 3) = (4, 0). The figure above shows exactly this translation.
Point P is at (-2, 6). Translate P by the vector (5, -8)ᵀ. Find the coordinates of the image point P'.
- New x = -2 + 5 = 3.
- New y = 6 + (-8) = -2.
- P' = (3, -2).
Answer: (3, -2)
- A whole shape is translated by (0, 0)ᵀ. What happens to it?
- How do you know every vertex of a translated shape moves by the SAME vector?
3. Adding and subtracting vectors
AbstractAdding two vectors combines two movements into one: walk vector a, then walk vector b, and the total displacement is a + b, found by adding the matching components. Subtracting works the same way, component by component.
For a = (3, 2)ᵀ and b = (-1, 4)ᵀ: a + b = (3 + (-1), 2 + 4)ᵀ = (2, 6)ᵀ. a - b = (3 - (-1), 2 - 4)ᵀ = (4, -2)ᵀ. Notice a - b is NOT the same as b - a: b - a = (-1 - 3, 4 - 2)ᵀ = (-4, 2)ᵀ, the exact opposite vector.
a = (4, -5)ᵀ and b = (-2, 3)ᵀ. Find a + b and a - b.
- a + b = (4 + (-2), -5 + 3)ᵀ = (2, -2)ᵀ.
- a - b = (4 - (-2), -5 - 3)ᵀ = (6, -8)ᵀ.
Answer: a + b = (2, -2)ᵀ. a - b = (6, -8)ᵀ.
- Is a - b always the opposite of b - a? Why?
- If a + b = (0, 0)ᵀ, what must be true about a and b?
4. Scalar multiplication and combining vectors
AbstractMultiplying a vector by a scalar (an ordinary number) scales every component by that number. A vector 3a points the same direction as a but three times as far; -a points the exact opposite direction, the same distance. Combining scaling with addition/subtraction, such as 2a - b, means scale first, then add or subtract.
For a = (3, 2)ᵀ: 2a = (6, 4)ᵀ. -a = (-3, -2)ᵀ. For a = (3, 2)ᵀ and b = (-1, 4)ᵀ: 2a - b = (6, 4)ᵀ - (-1, 4)ᵀ = (6 - (-1), 4 - 4)ᵀ = (7, 0)ᵀ.
a = (2, -3)ᵀ and b = (1, 5)ᵀ. Find 3a + 2b.
- 3a = (3 x 2, 3 x -3)ᵀ = (6, -9)ᵀ.
- 2b = (2 x 1, 2 x 5)ᵀ = (2, 10)ᵀ.
- 3a + 2b = (6 + 2, -9 + 10)ᵀ = (8, 1)ᵀ.
Answer: (8, 1)ᵀ
- What does the vector -2a look like compared to a: same direction or opposite?
- In 2a - b, which operation happens first: the scaling or the subtracting?
Common misconceptions and how to address them
MisconceptionThe translation vector from A to B is found by (old) - (new), not (new) - (old).
Why it happens: Students get the subtraction order backwards, especially when coordinates are negative.
How to address it: Always write dx = (image x) - (original x) and dy = (image y) - (original y). Checking: applying the resulting vector back to the ORIGINAL point should land exactly on the image point.
MisconceptionMultiplying a vector by a negative scalar just makes the numbers negative, with no effect on direction.
Why it happens: Students focus on the arithmetic (flipping signs) without connecting it to what the vector represents.
How to address it: -a reverses the direction completely (walks the exact opposite way) while keeping the same distance. Show this on a diagram: a and -a point directly away from each other through the origin.
Misconceptiona - b means subtract in whichever order the letters happen to be written in the final combined expression, e.g. treating 2a - b as 2a - a - b.
Why it happens: Students mix up combining LIKE vectors (which does not apply here, a and b are different vectors) with ordinary algebraic simplification.
How to address it: a and b are different vectors; 2a - b means "double vector a, then subtract vector b", two separate operations on two different quantities, worked component by component, never combined into a single letter.
Guided practice (with answers)
1. Point M is at (0, 3). It is translated to M' at (4, -1). Write the translation vector.
Answer: (4, -4)ᵀ, because dx = 4 - 0 = 4 and dy = -1 - 3 = -4.
2. Point N is at (-5, -2). Translate N by the vector (3, 6)ᵀ. Find N'.
Answer: (-2, 4), because -5 + 3 = -2 and -2 + 6 = 4.
3. a = (2, 5)ᵀ and b = (6, -1)ᵀ. Find a + b.
Answer: (8, 4)ᵀ, because 2 + 6 = 8 and 5 + (-1) = 4.
4. a = (2, 5)ᵀ and b = (6, -1)ᵀ. Find a - b.
Answer: (-4, 6)ᵀ, because 2 - 6 = -4 and 5 - (-1) = 6.
5. a = (-2, 3)ᵀ. Find 4a.
Answer: (-8, 12)ᵀ, because 4 x -2 = -8 and 4 x 3 = 12.
Independent practice worksheets
Practise describing and applying translation vectors, and vector arithmetic, with computed, never-wrong answer keys.
Differentiation
- Write dx and dy as two separate subtraction sums on squared paper before combining them into a column vector, so the 'image minus original' order is never rushed.
- Use a coordinate grid for every early question, plotting both the original and image points, so the vector is always visually confirmed, not just calculated.
- Introduce vector addition with both vectors having all-positive components first, before mixing in negative components.
- Keep scalars to 2 and 3 initially, before introducing negative scalars.
- Ask students to find the vector that, when added to a given vector, produces the zero vector (0, 0)ᵀ, connecting to the idea of an inverse.
- Introduce a real-world multi-step journey (e.g. three legs of a delivery route) and ask for the single resultant vector.
- Investigate what 0a (the zero scalar) and 1a (the scalar 1) each produce, and why.
- Preview (without requiring calculation) that a vector's magnitude, its length, connects to Pythagoras' theorem, a natural bridge to this site's Pythagoras & trigonometry unit.
Assessment: exit ticket
A three-question exit ticket sampling describing a translation, applying a vector, and vector arithmetic.
1. Point X is at (1, -4). It is translated to X' at (-2, 1). Write the translation vector.
Answer: (-3, 5)ᵀ, because dx = -2 - 1 = -3 and dy = 1 - (-4) = 5.
2. a = (5, -2)ᵀ and b = (-3, 4)ᵀ. Find a - b.
Answer: (8, -6)ᵀ, because 5 - (-3) = 8 and -2 - 4 = -6.
3. a = (1, -1)ᵀ. Find -3a.
Answer: (-3, 3)ᵀ, because -3 x 1 = -3 and -3 x -1 = 3.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 describing and applying translations (sections 1-2), Lesson 2 vector addition and subtraction (section 3), Lesson 3 scalar multiplication and combined expressions (section 4) plus the exit ticket.
- This unit assumes comfort with coordinates in all four quadrants and simple translations (Grade 7 transformations). Revisit that first if plotting points, not the vector notation, is the sticking point.
- Curriculum note: this unit cites the DfE 'GCSE mathematics: subject content and assessment objectives' (2013), items 24-25 of the 'Geometry and measures' section's 'Vectors' sub-heading. Item 25's final clause, 'use vectors to construct geometric arguments and proofs', is bold/Higher-tier only and is deliberately NOT covered here or in the accompanying worksheets, which stay scoped to Foundation tier throughout.
- Language to repeat: in compact notation a column vector is (horizontal, vertical)ᵀ, always in that order; the translation vector between two points is always (image) - (original); scaling a vector multiplies EVERY component by the same scalar.
- Present and print both work: use the Print button for a clean handout, or project the translation and vector-addition diagrams and build the coordinate arithmetic with the class live.