Vectors

A vector is a quantity defined by both a magnitude (size) and a direction. Velocity, force, displacement, and acceleration are all vectors; by contrast, quantities like temperature, mass, and speed have magnitude only and are called scalars. Graphically, a vector is drawn as an arrow whose length represents its magnitude and whose orientation gives its direction.

Vectors are most useful in component form. A two-dimensional vector v can be written as ⟨x, y⟩, splitting it into horizontal and vertical parts. A vector with magnitude m at angle θ from the horizontal has components x = m cos θ and y = m sin θ, and the Pythagorean theorem recovers the magnitude from components: |v| = √(x² + y²). Components turn geometry into arithmetic — to add two vectors, add their components; to multiply by a scalar, multiply each component.

This component approach powers real problem solving. In physics, a projectile's motion separates cleanly into independent horizontal and vertical components; forces on an object are resolved into components before applying Newton's laws; and equilibrium means the vector sum of forces is zero. Higher-level applications include the dot product, matrix operations on vectors, and vector representations of physical systems in engineering dynamics.

Vectors show up across a wide range of exams. The ACT tests vector components and addition in coordinate geometry, the AP Physics 1 exam builds its entire kinematics and force framework on vector analysis, and the FE Civil exam uses vectors in both its mathematics (matrices and vectors) and dynamics sections.