Vectors Summary

Pre-Calculus · Axiom Academy

Everything from this unit, from component form to projections, pulled together into one map. A vector carries both magnitude and direction ; in component form those live in the two coordinates at once. Addition, subtraction, and scalar multiplication all act component-by-component — no special rules to memorize. The dot product collapses two vectors into a single number that encodes the angle between them. That one number tells you everything geometric: the angle , whether vectors are orthogonal ( ), and the projection of one vector onto another. A vector is described by how far it reaches horizontally ( a ) and vertically ( b ). The standard unit vectors and let you write the same vector as a sum. When to use: any time you need to compute — components turn geometry into arithmetic. Watch out for: the position vector from P to Q is , tip minus tail. Core Concept Vector Operations Add and subtract by matching coordinates; scale by multiplying every component, . Scaling stretches or flips a vector without changing the line it points along. When to use: combining forces, velocities, or displacements. Watch out for: a negative scalar reverses direction. Core Concept Magnitude & Unit Vectors Magnitude is the vector's length — the Pythagorean theorem on its components. Dividing by that length normalizes the vector: a unit vector keeps the direction but has length 1 . When to use: to peel direction apart from size. Watch out for: you can't normalize the zero vector.

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