Vectors and Three-Dimensional Geometry
Core concepts
You must handle vectors in column and i, j, k notation, addition and scalar multiplication, the scalar (dot) product, modulus, unit vectors, and the equation of a line in three dimensions. Know how to find the angle between two vectors, test perpendicularity, and find the point of intersection of two lines or establish that they are skew.
Traps
- The dot product gives a scalar, not a vector; mixing it up with the cross product causes errors.
- Perpendicularity is a·b = 0, not a + b = 0.
- Two lines may be parallel, intersecting, or skew. Show that no single parameter value satisfies all three coordinate equations for skew lines.
- When computing the angle between two lines, use the direction vectors, not position vectors.
Worked example
Find the angle between a = (1, 2, 2) and b = (2, −1, 0). a·b = 2 − 2 + 0 = 0, so the vectors are perpendicular and the angle is 90°. Rule of thumb: compute the dot product first; if it is zero, no further work is needed.