Parallel and perpendicular components

Parallel and perpendicular components: linear algebra practice

Decompose a vector relative to a line through the origin. The parallel part is an orthogonal projection; subtracting it gives the perpendicular residual. Questions ask you to compute both pieces, verify their defining properties, use the resulting right-triangle relationship and distinguish an arbitrary split from an orthogonal decomposition.

What you will practise

• Split a vector into parallel and perpendicular components
• Verify the residual and the Pythagorean length relationship
• Explain the uniqueness and sign behavior of the decomposition

How to use this questionset

Prerequisites: Vector projection, orthogonality, norms and subtraction. 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 reference direction is nonzero. A correct decomposition must both reconstruct the original vector and satisfy the parallel/perpendicular conditions. This decomposition is the geometric basis of point-to-line distances and later least-squares residuals. This microlearning belongs to Module 1, Vectors and Geometry, in the Complete Linear Algebra Course.