Vector Algebra and Three Dimensional Geometry
Vector Algebra and Three Dimensional Geometry covers vector operations and their geometric meaning, then lines and planes in space. Expect direction cosines, dot and cross products, projections, area and volume, the vector and Cartesian equations of a line and a plane, and the angle between them.
Core formulas
- |a| = √(x² + y² + z²); unit vector â = a/|a|; direction cosines satisfy l² + m² + n² = 1.
- a·b = |a||b|cos θ = x₁x₂ + y₁y₂ + z₁z₂; a·b = 0 means perpendicular. Projection of a on b is (a·b)/|b|.
- a × b = |a||b|sin θ n̂; a × b = 0 means parallel. |a × b| is the area of the parallelogram, and half of it the area of the triangle.
- Scalar triple product [a b c] = a·(b × c) is the volume of the parallelepiped; [a b c] = 0 means coplanar.
- Line: r = a + λb, or (x − x₁)/l = (y − y₁)/m = (z − z₁)/n. Plane: r·n = d, or lx + my + nz = d.
Worked example
Find the area of the triangle with vertices A(1, 1, 1), B(1, 2, 3), C(2, 3, 1). AB = (0, 1, 2) and AC = (1, 2, 0). Then AB × AC = (1·0 − 2·2, 2·1 − 0·0, 0·2 − 1·1) = (−4, 2, −1). Its magnitude is √(16 + 4 + 1) = √21, so the area is √21/2 square units.
Traps
- Using a·b = 0 to test parallelism; that test belongs to a × b = 0.
- Quoting the cross product vector as the area instead of its magnitude.
- Mixing direction ratios with direction cosines.
- Dividing by |a| when the projection of a on b requires division by |b|.