Pure 2 vectors
Use coordinates and vectors to describe three-dimensional geometry and motion.
Begin with coordinates and distance. Then work through vector operations, geometry, proofs, forces and motion, and finish with mixed practice.
- Coordinates and distance in 3DRead three-dimensional coordinates and calculate distances using Pythagoras. Find unknown coordinates, handle negative components, locate midpoints and distinguish a space diagonal from a surface route.
- Vector components and displacementUse three-component vectors, distinguish position from displacement, combine journeys head to tail and solve vector equations component by component.
- Magnitudes, unit vectors and parallelismCalculate vector lengths, form unit vectors in a specified direction, solve magnitude conditions and test parallelism or collinearity without dividing by zero components.
- Angles with axes and planesFind the angles a 3D vector makes with positive coordinate axes and distinguish them from its inclination to a coordinate plane. Handle negative components, vertical directions and the zero vector.
- Triangles and areas in 3DUse vector displacements to find triangle side lengths, angles and areas in three dimensions. Choose the correct vertex, test right angles and split a planar quadrilateral into triangles.
- Parallelograms, ratios and reflectionsFind missing parallelogram vertices, divide a segment in a specified ratio, locate external points and reflect a point in a line using its perpendicular foot.
- Vector proofs in three dimensionsBuild vector proofs using independent directions, compare every component, verify line intersections and prove diagonal and midpoint results in three-dimensional shapes.
- Forces and equilibrium in 3DAdd force vectors, apply Newton’s second law component by component, find an equilibrant and include weight with the correct vertical sign.
- Motion with 3D vectorsUse constant vector acceleration to find velocity and position, distinguish distance from displacement and use vertical velocity to decide ascent or descent.
- 3D vectors: mixed practiceTwenty original mixed vector questions with separate hints and worked solutions, covering 3D geometry, ratios, proofs, forces and motion.