Vectors

Maths · Geometry and measures. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A vector has a size and a direction. A column vector gives a movement: the top number is the move right (negative means left) and the bottom number is the move up (negative means down). A translation is described by one column vector.

Vectors can also be written with letters, such as a and b. To get from one point to another, follow any route along vectors you know and add them. Going against the arrow changes the sign, so the vector from B to A is the negative of the vector from A to B.

In proofs, the main idea is that two vectors are parallel when one is a multiple of the other. If they also share a point, the points lie on one straight line.

Rules to remember

Translation vector = (new x − old x, new y − old y), top entry first.
Add or subtract column vectors entry by entry; multiplying by a number (a scalar) multiplies each entry.
Vector AB = b − a, where a and b are the vectors from O to A and from O to B.
Vector BA = −(vector AB).
The midpoint M of AB has vector OM = ½(a + b).
Two vectors are parallel when one is a multiple of the other, e.g. 6a − 3b = 3(2a − b).
If vector PQ and vector QR are multiples of each other, then P, Q and R lie on one straight line.

Common mistake

Writing vector AB as a − b. To go from A to B you travel back along a and then along b, so vector AB = −a + b = b − a.

Worked example

Vector OA = 2a and vector OB = 3b. M is the midpoint of AB. Find vector OM in terms of a and b.

Vector AB = vector AO + vector OB = −2a + 3b.
Vector AM = ½ × vector AB = −a + (3/2)b.
Vector OM = vector OA + vector AM = 2a − a + (3/2)b = a + (3/2)b.

Answer: Vector OM = a + (3/2)b

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