Vectors
What this domain covers
Vectors in 8227 are examined in two and three dimensions: component notation, magnitude and unit vectors, addition and subtraction, the scalar product and the angle between vectors, plus simple geometric applications such as midpoints, collinearity and ratio division.
Core rules
- Component form: a = a₁i + a₂j + a₃k. Addition and subtraction work componentwise; λa scales every component.
- Magnitude: |a| = √(a₁² + a₂² + a₃²); unit vector â = a/|a|.
- Scalar product: a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ.
- Perpendicular vectors have a · b = 0; parallel vectors are scalar multiples of one another.
- Vector AB = b − a, the position vector of B minus that of A, and |AB| is the distance AB.
- Collinearity: A, B, C lie on a line when AB and BC are parallel and share the point B, so one is a scalar multiple of the other.
Worked example
a = 2i − j + 3k and b = i + 4j − k. Find the angle between them. a · b = (2)(1) + (−1)(4) + (3)(−1) = 2 − 4 − 3 = −5. Also |a| = √(4 + 1 + 9) = √14 and |b| = √(1 + 16 + 1) = √18. So cos θ = −5/(√14 · √18) = −5/√252 ≈ −0.3149, giving θ ≈ 108.4°. The negative scalar product already signals an obtuse angle, which is a useful check.
Traps
- Writing AB = a − b instead of b − a.
- Squaring the magnitude rather than square-rooting the sum of squares.
- Assuming i · j = 1; the scalar product of perpendicular base vectors is 0.
- Forming a unit vector by dividing by one component instead of by the magnitude.
- Proving two vectors are parallel but forgetting to state the shared point needed for collinearity.