Vectors
Vectors in SPM Additional Mathematics are two-dimensional. The exam tests magnitude, unit vectors, addition and subtraction, multiplication by a scalar, position vectors and collinearity.
Core rules
A vector may be written as xi + yj or as a column vector. Its magnitude is |xi + yj| = √(x² + y²). The unit vector in the same direction is the vector divided by its magnitude. For points A and B with position vectors a and b, the vector AB = b − a. Two vectors are parallel if one is a scalar multiple of the other; points A, B and C are collinear if AB = kBC for some scalar k and the two vectors share a point.
Worked example
Given A(1, 2), B(4, 6) and C(7, 10): AB = 3i + 4j, so |AB| = √(9 + 16) = 5 and the unit vector is (3/5)i + (4/5)j. Also BC = 3i + 4j = AB, and both vectors contain B, so A, B and C are collinear.
Traps
Writing BA when AB is required; giving a negative magnitude; claiming vectors are parallel without exhibiting a scalar multiple; forgetting the shared point in a collinearity proof.