Vector Algebra and Three-Dimensional Geometry
Vector Algebra and Three-Dimensional Geometry
The exam tests vector addition, dot and cross products, projections, direction cosines and ratios, equations of lines and planes, angles and shortest distances.
Core rules
- a·b = |a||b|cosθ and a×b = |a||b|sinθ n̂. Perpendicular vectors have a·b = 0; parallel vectors have a×b = 0.
- Projection of a on b = (a·b)/|b|; the unit vector along a is a/|a|.
- Direction cosines l, m, n satisfy l² + m² + n² = 1; direction ratios are proportional numbers, not necessarily cosines.
- Line: r = a + λb. Plane: r·n = d, or ax + by + cz = d with normal n = ai + bj + ck.
- Distance of (x₀,y₀,z₀) from ax + by + cz + d = 0 is |ax₀ + by₀ + cz₀ + d|/√(a² + b² + c²).
Worked example
Find the angle between a = i + j and b = j + k.
a·b = (1)(0) + (1)(1) + (0)(1) = 1. |a| = √2 and |b| = √2. Therefore cosθ = 1/(√2·√2) = 1/2, so θ = π/3, i.e. 60°.
Traps
- Reversing the order in a cross product; b×a = −a×b.
- Calling direction ratios direction cosines without normalising.
- Using cosθ instead of sinθ for the angle between a line and a plane.
- Forgetting the absolute value in point-to-plane distance.
- Mixing the line direction vector with the plane normal vector.