A column vector describes a movement using two numbers written vertically: the top number gives the horizontal shift (positive = right, negative = left) and the bottom number gives the vertical shift (positive = up, negative = down). Column vectors are the standard notation for describing translations in KS3 geometry.
What is a column vector?
A column vector is written as two numbers stacked in a bracket:
$$\mathbf{v} = \begin{pmatrix} a \ b \end{pmatrix}$$
where a is the horizontal component and b is the vertical component.
Examples:
| Vector | Meaning |
|---|---|
| (3 over 2) | 3 right, 2 up |
| (−4 over 1) | 4 left, 1 up |
| (5 over −3) | 5 right, 3 down |
| (0 over −2) | no horizontal movement, 2 down |
| (−1 over −5) | 1 left, 5 down |
In a diagram, draw an arrow from the starting point in the direction described. The length and direction of the arrow represent the vector.
How do you use a column vector to describe a translation?
To describe a translation using a column vector:
- Pick any point on the original shape (the object).
- Find the corresponding point on the image (the translated shape).
- Count how many squares right or left (the horizontal component, positive right).
- Count how many squares up or down (the vertical component, positive up).
- Write the result as a column vector.
Worked example: A point moves from (2, 5) to (7, 3).
- Horizontal: 7 − 2 = +5 (5 to the right)
- Vertical: 3 − 5 = −2 (2 downwards)
- Translation vector: (5 over −2)
To apply the translation, add the vector components to each coordinate of the original shape.
How do you add and subtract column vectors?
Adding vectors: Add the top numbers together and the bottom numbers together.
$$\begin{pmatrix} 3 \ 1 \end{pmatrix} + \begin{pmatrix} 2 \ -4 \end{pmatrix} = \begin{pmatrix} 3+2 \ 1+(-4) \end{pmatrix} = \begin{pmatrix} 5 \ -3 \end{pmatrix}$$
Subtracting vectors: Subtract the top numbers and the bottom numbers.
$$\begin{pmatrix} 6 \ -1 \end{pmatrix} - \begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} 6-2 \ -1-3 \end{pmatrix} = \begin{pmatrix} 4 \ -4 \end{pmatrix}$$
Multiplying by a scalar (a number): Multiply both components by the scalar.
$$3 \times \begin{pmatrix} 2 \ -1 \end{pmatrix} = \begin{pmatrix} 6 \ -3 \end{pmatrix}$$
What is the negative of a vector?
The negative vector reverses the direction of movement. To negate a vector, change the sign of both components.
$$-\begin{pmatrix} 4 \ -3 \end{pmatrix} = \begin{pmatrix} -4 \ 3 \end{pmatrix}$$
If vector a translates a shape from A to B, then −a translates it from B back to A. This is useful when you need to describe the reverse journey.
Worked example: applying a translation to a shape
Triangle T has vertices at A(1, 2), B(3, 2), and C(2, 4). Apply the translation (4 over −3).
Add the vector to each vertex:
- A(1, 2) → (1 + 4, 2 − 3) = A'(5, −1)
- B(3, 2) → (3 + 4, 2 − 3) = B'(7, −1)
- C(2, 4) → (2 + 4, 4 − 3) = C'(6, 1)
Triangle T' has vertices at A'(5, −1), B'(7, −1), C'(6, 1).
Check: the shape, size, and orientation of T' are identical to T — translations never change these. ✓
What mistakes do students commonly make?
- Mixing up horizontal and vertical. The top number is always horizontal (across); the bottom is always vertical (up/down). Students sometimes swap them.
- Getting the sign of the vertical wrong. "Up" is positive on a standard grid; "down" is negative. A common error is to make downward movements positive because they feel lower.
- Not applying the vector to every vertex. In a translation, every point of the shape moves by the same vector. If you only move some vertices, the shape will distort.
- Confusing vector notation with coordinates. A column vector describes a movement; a coordinate describes a position. (3, 4) as a coordinate means the point three right and four up from the origin; as a vector it means "move 3 right and 4 up from wherever you start".
Frequently asked questions
How is a column vector different from a coordinate?
A coordinate gives a fixed position relative to the origin (0, 0). A column vector gives a displacement — a change in position. The vector (3 over −2) can be applied from any starting point; the coordinate (3, −2) names a specific point on the grid.
Can a column vector have a component of zero?
Yes. A vector (0 over 5) means "no horizontal movement, 5 upward". A vector (3 over 0) means "3 to the right, no vertical movement". Both are valid column vectors.
What happens to the properties of a shape after a translation?
A translation preserves the shape's size, angles, and orientation exactly. The image is congruent to the object — it is simply in a different position. No rotation or reflection occurs; every point moves by exactly the same vector.
How do you find the vector that takes shape A to shape B?
Choose any vertex on shape A and find its image on shape B. The vector from A's vertex to B's vertex is the translation vector: vector = (x_B − x_A) over (y_B − y_A). Check with a second vertex pair to confirm the same vector applies.
For guided practice on vectors and transformations at KS3 with instant Socratic feedback, visit aitutors.me.