Vectors

A vector has a size and a direction. A column vector shows the movement across and then up; here it's written as (3 over −2): 3 right and 2 down.

Vectors are named with a bold lowercase letter, such as a, or by their start and end points, such as AB for the vector from A to B. (In the exam, underline a letter or draw an arrow over AB.)

To draw a vector, start anywhere: move across by the top number, then up by the bottom one, and draw an arrow.

Example: A is (1, 2) and B is (4, −1). Write AB as a column vector.

1. Across: 4 − 1 = 3
2. Up: −1 − 2 = −3
3. AB = (3 over −3)

• To add or subtract, add or subtract the top numbers, then the bottom numbers.
• To multiply by a number (a scalar), multiply both numbers.

Adding vectors means following one, then the other. Multiplying by 2 doubles the length, and multiplying by −1 reverses the direction.

Example: a = (2 over 5) and b = (−3 over 1). Work out a + b, a − b, 3a and 2a − b.

1. a + b = (2 + −3 over 5 + 1) = (−1 over 6)
2. a − b = (2 − −3 over 5 − 1) = (5 over 4)
3. 3a = (6 over 15)
4. 2a − b = (4 − −3 over 10 − 1) = (7 over 9)

To find a vector between two points, follow a route made of vectors you know. Going backwards along a vector changes its sign.

Example: OA = a and OB = b. Find AB.

1. From A to B, go back from A to O, then from O to B
2. AB = −a + b = b − a

Example: M is the midpoint of AB. Find OM.

1. OM = OA + AM, and AM is half of AB
2. OM = a + ½(b − a)
3. = ½a + ½b

A point that divides a line in a ratio is a fraction of the way along.

Example: OA = a and OB = b. P is on AB with AP : PB = 1 : 2. Find OP.

1. P is ⅓ of the way from A to B, so AP = ⅓AB = ⅓(b − a)
2. OP = OA + AP = a + ⅓(b − a)
3. = a + ⅓b − ⅓a
4. = ⅔a + ⅓b

• Two vectors are parallel if one is a multiple of the other: 3a − 6b = 3(a − 2b) is parallel to a − 2b.
• Three points are collinear (on one straight line) if the vectors between them are parallel and share a point.

Example: OA = a, OB = b and OC = 3b − 2a. Prove that A, B and C are collinear.

1. AB = b − a
2. AC = OC − OA = 3b − 2a − a = 3b − 3a = 3(b − a)
3. AC = 3AB, so AB and AC are parallel
4. They share the point A, so A, B and C are collinear