Vectors
What the domain tests
Vectors tests representation, addition, scalar multiplication, magnitude and the scalar product, with applications to geometry and collinearity. MSCE Additional Mathematics questions often use position vectors and require a proof, so the direction of each vector and the order of subtraction matter.
Core rules
- A vector has magnitude and direction. In column form a=(x,y); magnitude |a|=√(x^2+y^2). The unit vector in the direction of a is a/|a|.
- Addition is componentwise. If A and B have position vectors a and b, then AB=b-a (head minus tail).
- Scalar product: a·b=|a||b|cosθ=x1x2+y1y2. If a·b=0 and both vectors are non-zero, they are perpendicular.
- Points A, B and C are collinear if AB=kBC for some scalar k, or if AB and AC are parallel.
- A line has equation r=a+tb, where a is a position vector on the line and b is a direction vector.
Worked example
Given a=(2,1) and b=(1,3), find the angle between them. a·b=2(1)+1(3)=5. |a|=√5 and |b|=√10. Hence cosθ=5/(√5√10)=5/√50=1/√2, so θ=45°.
Traps
Subtracting position vectors in the wrong order; confusing a vector with its magnitude; using the scalar product formula without the cosine definition; proving collinearity from equal magnitudes rather than parallel directions.