Further Work with Vectors (MEX-V1)

3 dot points across 3 inquiry questions, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How are vectors in three dimensions described, how do we find lines and spheres, and how does the scalar product extend to space?

Represent three-dimensional vectors in component form, compute the scalar product and magnitude, and find vector equations of lines and the equation of a sphere

3D vectors. Component form with i, j, k, magnitude as a space diagonal, the scalar product and angle in space, parametric vector equations of lines, intersecting versus skew lines, and the equation of a sphere, built up stage by stage with verified worked examples and diagrams.

14 min read →

How can vector methods prove classical geometric results such as concurrency, midpoint theorems and properties of parallelograms more cleanly than coordinate geometry?

Prove geometric results using vectors, including properties of triangles, parallelograms and the diagonals of quadrilaterals, by expressing points as position vectors

Vector geometric proofs. Position vectors and the four-step tactic, the parallel, collinear and perpendicular tests, the midpoint and section formulas, proofs of classical 2D results, and proofs about 3D figures such as the cube, with verified worked examples and diagrams.

14 min read →

How do we resolve one vector along the direction of another, and what do the scalar and vector projections measure geometrically?

Compute the projection of one vector onto another, distinguishing the scalar projection from the vector projection, and resolve a vector into parallel and perpendicular components

Vector projection. The scalar projection as a signed length, the vector projection formula, the parallel and perpendicular split in three dimensions, and the distance from a point to a line built up stage by stage, with verified worked examples and diagrams.

13 min read →