Vectors in three dimensions.
Develop accurate geometric and algebraic methods for vectors, lines, planes, distances and systems in three dimensions.
Work through the topic
Learn the method, follow the examples, then try the worksheet.
- 01
Three-dimensional coordinates, magnitude and unit vectors
Locate points in space, calculate vector lengths and construct unit directions.
- 02
Vector operations
Combine spatial displacements and locate points using vector equations.
- 03
Dot product and projection
Use dot products for angles, perpendicularity and vector resolution.
- 04
Cross product and determinant calculations
Calculate oriented cross products and use determinant checks.
- 05
Geometric areas and perpendicular vectors
Find geometric areas and normal vectors using a spanning pair.
- 06
Vector/Cartesian equations of lines
Write vector and Cartesian line equations and interpret parameter restrictions.
- 07
Plane equations and normal vectors
Build plane equations from normals, points and spanning directions.
- 08
Line/plane intersections and angles
Solve intersections and distinguish vector, line-plane and plane-plane angles.
- 09
Closest points and distances including skew lines
Use perpendicularity to find closest points and distances, including skew lines.
- 10
Systems of three linear equations
Solve three linear equations and interpret unique, inconsistent and parameterised solutions.
- 11
Row operations and geometric interpretation of solution sets
Use row operations accurately and interpret pivots and free parameters.
Topic review
Revisit the key ideas, then choose a practice paper.