Directed segments, vectors and translation vectors
Draw vectors from tail to head and read them as column vectors, find equal and opposite vectors, and translate a shape by a vector.
📖 Start here: read the steps of the “Lesson” in order (1, 2, 3 …) and answer each question. Then continue with “Practice”.
- UK: Year 10
- France: grade 9
- Germany: grade 7
- Iran: grade 7
- Syria: grade 8
- Ukraine: grade 9
- Russia: grade 9
Guide
A line segment with a direction is a vector. It starts at its tail and ends at its head, and the arrowhead is always at the head. The vector AB starts at A and ends at B. The vector BA is the same segment, the other way round.
We write a vector as a column vector. The top number is the move across: right, or left if it is negative. The bottom number is the move up, or down if it is negative. So 4−3 means 4 right and 3 down.
Two vectors are equal when they have the same length and the same direction, wherever they are on the grid. The negative of a vector has the same length but points the opposite way: the negative of 2−3 is −23. In a translation, every point of the shape moves by the same vector, the translation vector. Head = tail + vector.