Dot products and angles

Connect vector algebra to geometry through dot products, orthogonality and angles between vectors. Practice component calculations, scalar identities, parameter conditions and angle recovery, then use length information and feasibility checks to reason about vectors without relying on a diagram.

Single-select and select-all questions progress toward multi-step deductions with detailed explanations. The set emphasizes that a dot product is a scalar, that angle formulas require two nonzero vectors, and that a computed cosine must lie between negative one and one. Projection and work are reserved for the following topic, keeping this practice focused on dot products and angles.