Vectors and three-dimensional geometry
Vectors and three-dimensional geometry tests vector arithmetic in column and i, j, k notation, magnitude and unit vectors, the scalar product, angles between vectors, and the vector equation of a line in three dimensions, including intersections and the classification of line pairs.
Vector arithmetic and magnitude
The magnitude of a = (x, y, z) is |a| = √(x² + y² + z²), and the unit vector in its direction is a/|a|. Position vectors locate points relative to an origin, and the vector from A to B is the position vector of B minus that of A.
Scalar product and angle
The scalar product is a·b = |a||b| cos θ, and in components a·b = x₁x₂ + y₁y₂ + z₁z₂. Hence cos θ = (a·b)/(|a||b|). Perpendicular vectors have zero scalar product. The scalar product is commutative and distributes over addition.
Lines in three dimensions
A line through point a with direction b has equation r = a + λb. Two lines may be parallel (direction vectors are multiples), intersecting (a unique common point exists), or skew (neither parallel nor intersecting, which is possible only in three dimensions). To test intersection, equate components and solve for the two parameters, then check the third component for consistency.
Worked example
Find the angle between a = (1, 2, 2) and b = (2, −1, 2).
a·b = 2 − 2 + 4 = 4. |a| = √(1 + 4 + 4) = 3 and |b| = √(4 + 1 + 4) = 3. So cos θ = 4/9, giving θ ≈ 63.6°.
Traps that cost marks
- Using different parameters λ and μ for a single line, or the same parameter for two different lines.
- Concluding two lines intersect after checking only two components.
- Confusing a direction vector with a position vector in the line equation.
- Forgetting the absolute value or the vector magnitudes when computing an angle.