AP Precalculus · Lesson 4 of 4
Parametric functions, vectors, and matrices
Describe motion with a parameter, combine vectors by components, and use matrices to represent linear relationships.
Parametric functions and vectors
If x and y both depend on t, the path is a parametric curve. Eliminating t can produce a Cartesian equation, but the allowed values of t still restrict which part of that equation is traced. The direction of motion comes from how t increases.
A vector in the plane has components. Adding vectors adds the corresponding components. The magnitude of ⟨a, b⟩ is √(a² + b²), which is a length. A velocity vector’s components are horizontal and vertical rates, and its magnitude is speed.
Matrices
A matrix can store the coefficients of a linear system. Multiplying a matrix by a vector applies the linear transformation that matrix represents. The order of matrix multiplication matters: AB and BA are not generally equal, and one of them may not even be defined if the sizes do not match.
The identity matrix leaves a compatible vector unchanged. An inverse, when it exists, undoes the transformation. A system with no unique solution does not have an invertible coefficient matrix.
Worked example
A particle has position ⟨3t, t²⟩ at time t. What is its position and speed at t = 2?
- The position is ⟨6, 4⟩.
- The velocity is the derivative ⟨3, 2t⟩, so at t = 2 it is ⟨3, 4⟩.
- The speed is √(3² + 4²) = 5.
Why this works. Differentiate each component to get velocity, then take the magnitude to get speed.
Check your understanding
- State the interval of the parameter, not only the Cartesian path.
- Add vectors component by component and compute magnitude with the Pythagorean relation.
- Check that matrix dimensions match before multiplying, and keep the multiplication order.