Vectors
Vectors
Vectors appears in both the pure and mechanics halves of this paper: column and i, j notation, magnitude and direction, position vectors, the scalar product and the geometric consequences of parallelism and perpendicularity. Mechanics questions use the same algebra for displacements and velocities.
Basics
A vector has magnitude and direction; a scalar has magnitude only. Magnitude of ai + bj is √(a² + b²). The unit vector in that direction is (ai + bj)/√(a² + b²). Position vector →AB = b − a, so the length AB = |b − a|. Two vectors are parallel when one is a scalar multiple of the other, and points A, B, C are collinear when →AB is a multiple of →AC and they share a point.
Scalar product
a · b = |a||b| cos θ = a₁b₁ + a₂b₂. Consequences: a · b = 0 exactly when a and b are perpendicular (for non-zero vectors); cos θ = (a · b)/(|a||b|) gives the angle between them. Remember the dot product produces a scalar, never a vector.
Worked example
a = 3i − j and b = i + 2j. Find the angle between them. a · b = (3)(1) + (−1)(2) = 1. |a| = √(9 + 1) = √10 and |b| = √(1 + 4) = √5. So cos θ = 1/(√10 × √5) = 1/√50 ≈ 0.1414, giving θ ≈ 81.9°. Since the dot product is small and positive, the angle should be just under 90°, which it is.
Traps that cost marks
- Using →BA instead of →AB and obtaining the negative displacement.
- Forgetting that perpendicularity requires a · b = 0, not |a| = |b|.
- Mixing signs in the second component when both vectors have negative entries.
- Reporting the angle in degrees when the question works in radians.
- Proving collinearity without stating that the vectors share a common point.