Projection, decomposition and work

Build geometric understanding of vector projection, orthogonal decomposition and mechanical work in this precalculus practice set. Start by identifying parallel components, then calculate projections and verify perpendicular residuals. Later problems connect dot products to force, displacement, units and closest points on a line.

Single-select and multiselect questions include common sign and denominator errors, zero-vector restrictions and short justifications. Work each problem on paper before checking the choices; detailed explanations show the method, an answer check and the misconception behind competing answers. Familiarity with two-dimensional vectors and the dot product is recommended. No calculus is required.