Vectors: Dot Product, Cross Product & Geometric Applications
Vectors are tested extensively on IB Math AA HL and AI HL papers. The dot product of vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) is a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cosθ, where θ is the angle between them. If a·b = 0, the vectors are perpendicular. The cross product a × b produces a vector perpendicular to both a and b, with magnitude |a||b|sinθ. It is calculated using a 3×3 determinant expansion. The cross product is used to find the normal to a plane and to compute area of a parallelogram (|a × b|). Vector equation of a line: r = a + λd, where a is a position vector of a point on the line and d is the direction vector. Two lines are parallel if their direction vectors are scalar multiples; they intersect if there exists λ and μ satisfying the equations simultaneously; they are skew if they don't intersect and aren't parallel. Equation of a plane: r·n = a·n, where n is the normal vector. Alternatively, in Cartesian form: nx(x - a₁) + ny(y - a₂) + nz(z - a₃) = 0.