Vectors · Year 12
In Year 12 you learn: vector geometry proofs.
A vector has both size (magnitude) and direction. Column vectors describe moves, such as (3, −2) meaning 3 right and 2 down, and bold letters stand for vectors in geometric proofs about midpoints, parallel lines and straight lines.
Key facts
- Add or subtract vectors by adding or subtracting matching components.
- Multiplying by a number k scales the length; k < 0 reverses the direction.
- AB = b − a, where a and b are the position vectors of A and B; BA = −AB.
- Parallel vectors are multiples of each other; if AB = k × BC, then A, B and C lie on a straight line.
Worked examples
Example 1
a = (3, −1) and b = (−2, 4) are column vectors (across, up). Find 2a + b.
Answer: 2a + b = (4, 2)
Example 2
OA = a and OB = b. M is the midpoint of AB. Show that OM = ½(a + b).
Answer: OM = a + ½(b − a) = ½a + ½b = ½(a + b)