Adding vectors
To add two vectors we join them head to tail. If AB = a and BC = b, then the resultant is AC = a + b. This is the triangle law. When two vectors start from the same point, the parallelogram law gives the resultant as the diagonal of the parallelogram formed by a and b. Vector addition is commutative: a + b = b + a.
Subtracting vectors
Subtraction is defined as adding the negative vector: a − b = a + (−b). A useful result on any diagram is that for points O, A, B, AB = OB − OA, since AB = AO + OB = −OA + OB.
Key formula
Triangle law: AB + BC = AC. Subtraction: a − b = a + (−b). Using an origin O: AB = OB − OA.
Worked example
In triangle OAB, OA = a and OB = b. M is the midpoint of AB. Find OM in terms of a and b. First AB = OB − OA = b − a. Then AM = ½AB = ½(b − a). So OM = OA + AM = a + ½(b − a) = ½a + ½b = ½(a + b).
Working through several points
Many problems join more than two vectors. Always travel head to tail: AB + BC + CD = AD, since the intermediate points cancel. If you reach the starting point again, the total is the zero vector, e.g. AB + BC + CA = 0. When ratios divide a line, write the point as a start vector plus a fraction of the segment: if P divides AB in the ratio m:n, then OP = OA + [m/(m+n)]AB. These techniques let you express any point on a diagram in terms of two base vectors a and b, which is the heart of most examination questions.
Remember
- Head-to-tail joining gives the resultant.
- a − b means a + (−b).
- AB = OB − OA (final minus initial).