Vector projection

Vector projection: linear algebra practice

Project a vector orthogonally onto the line generated by a nonzero direction. Questions build the formula from two requirements: the answer must lie on the line, and the remaining displacement must be perpendicular to it. Numerical examples and error diagnoses distinguish a projection vector from related scalar quantities.

What you will practise

• Derive and calculate an orthogonal vector projection onto a line
• Check parallelism and an orthogonal residual
• Distinguish the projected vector, its coefficient and signed scalar projection

How to use this questionset

Prerequisites: Dot products, norms, scalar projection and orthogonality. The questions progress from direct interpretation to application and reasoning. Work out an answer before comparing the choices; for multi-select questions, evaluate every statement independently. Worked explanations show the reasoning and address relevant misconceptions, so an incorrect attempt can guide your next review.

The direction vector must be nonzero; normalization is optional when the squared-norm denominator is used correctly. This line-projection construction supports perpendicular decomposition and later least-squares geometry. This microlearning belongs to Module 1, Vectors and Geometry, in the Complete Linear Algebra Course.