A translation maps the point A(4, −2) onto B(1, 5). Find the image of C(0, 3) under the same translation.
- Translation vector = B − A = (1 − 4, 5 − (−2)) = (−3, 7).
- Add the vector to C: (0 + (−3), 3 + 7).
- Image = (−3, 10).
Answer: (−3, 10)
Vectors in two dimensions
Maths · Transformations and vectors. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
A vector has a size and a direction. A column vector has two numbers, one above the other. The top number is the movement in the x direction: positive is right, negative is left. The bottom number is the movement in the y direction: positive is up, negative is down. In these notes a column vector is written as (x, y), top number first.
A translation moves every point of a shape by the same vector. To find the vector, subtract: image minus object.
To add or subtract vectors, work with the top numbers and the bottom numbers separately. To multiply a vector by a number (a scalar), multiply both parts by that number.
Rules to remember
- Translation vector = coordinates of image − coordinates of object
- Add: (a, b) + (c, d) = (a + c, b + d)
- Subtract: (a, b) − (c, d) = (a − c, b − d)
- Multiply by a scalar: k(a, b) = (ka, kb)
- Journeys join up: vector AB + vector BC = vector AC
- The vector −a has the same length as a but the opposite direction
- Left and down are negative: 6 left and 2 down is (−6, −2)
Common mistake
Students subtract the wrong way round and get every sign wrong. The vector from A to B is always B minus A: end minus start.
Worked example
Ask about this
Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.