4.9 Vector-Valued Functions

A vector-valued function turns parametric motion into vectors. The position vector $p(t) = \langle x(t), y(t) \rangle$ tells you where a particle is, and the velocity vector $v(t) = \langle x'(t), y'(t) \rangle$ tells you how fast and which way it is moving.

Why This Matters for the AP Precalculus Exam

This topic lives in Unit 4, which is not assessed on the AP Precalculus exam. The exam tests Units 1, 2, and 3. Schools may still teach this topic because it builds directly into calculus, physics, and computer graphics, where motion gets broken into independent horizontal and vertical parts.

What you build here is still useful thinking. Vector-valued functions connect three earlier ideas: parametric functions, planar motion, and vectors. Getting comfortable with position and velocity vectors now makes calculus topics like motion in the plane feel familiar later. If your class includes Unit 4, expect to use a graphing calculator to plot parametric curves and analyze motion.

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Key Takeaways

• A position vector $p(t) = x(t)\vec{i} + y(t)\vec{j}$ or $p(t) = \langle x(t), y(t) \rangle$ gives a particle's location at time $t$.
• The magnitude $|p(t)|$ gives the particle's distance from the origin.
• The velocity vector $v(t) = \langle x'(t), y'(t) \rangle$ describes how the particle moves at time $t$.
• The sign of $x'(t)$ tells you right (positive) or left (negative); the sign of $y'(t)$ tells you up (positive) or down (negative).
• The magnitude of the velocity vector gives the particle's speed.
• Position and velocity are two different vectors. Do not mix up where something is with how it is moving.

Position Vector

The position of a particle moving in a two-dimensional plane can be written as a vector-valued function:

$p(t) = x(t)\vec{i} + y(t)\vec{j}$

Here $x(t)$ and $y(t)$ are the coordinates of the particle at time $t$, and $\vec{i}$ and $\vec{j}$ are the unit vectors in the x and y directions. You can also write the same position vector in component form:

$p(t) = \langle x(t), y(t) \rangle$

This is just the parametric function $f(t) = (x(t), y(t))$ written as a vector. Same information, new notation.

The magnitude of the position vector at time $t$, written $|p(t)|$, gives the distance of the particle from the origin (0, 0):

$|p(t)| = \sqrt{x(t)^2 + y(t)^2}$

Velocity Vector

The velocity of a particle moving in a plane is given by a separate vector-valued function:

$v(t) = \langle x'(t), y'(t) \rangle$

The components $x'(t)$ and $y'(t)$ are the horizontal and vertical velocities of the particle.

At any time $t$, the sign of each component tells you direction:

• If $x'(t) > 0$, the particle is moving right. If $x'(t) < 0$, it is moving left.
• If $y'(t) > 0$, the particle is moving up. If $y'(t) < 0$, it is moving down.

The magnitude of the velocity vector gives the speed of the particle:

$|v(t)| = \sqrt{x'(t)^2 + y'(t)^2}$

Speed is how fast the particle moves, with no direction attached. You can read it as the distance the particle would cover per unit of time if it kept that velocity.

How to Use This on the AP Precalculus Exam

Unit 4 is not on the AP Precalculus exam, so you will not be scored on vector-valued functions. Use these steps as practice for clear reasoning and for calculus prep if your class covers the unit.

Problem Solving

• Read whether the problem gives you position or velocity. Position uses $x(t)$ and $y(t)$; velocity uses $x'(t)$ and $y'(t)$.
• To find distance from the origin, take the magnitude of the position vector.
• To find speed, take the magnitude of the velocity vector.
• To describe direction of motion, check the signs of $x'(t)$ and $y'(t)$ separately. Horizontal and vertical motion are independent.
• Keep your notation consistent. Decide whether you are using $\langle x, y \rangle$ form or $\vec{i}, \vec{j}$ form and stick with it within a problem.

Common Trap

Plugging a time value into the position function when the question asks about speed or direction. Speed and direction come from the velocity vector $v(t) = \langle x'(t), y'(t) \rangle$, not from the position vector.

Common Misconceptions

• Position and velocity are not the same vector. $p(t)$ tells you where the particle is; $v(t)$ tells you how it is moving. A particle can be far from the origin but moving slowly, or close to the origin but moving fast.
• Velocity is not just one number. It has horizontal and vertical components. Speed is the single number you get from the magnitude of the velocity vector.
• The sign of a velocity component tells direction, not size. A velocity of $-5$ in the x direction means moving left at speed 5, not moving slowly.
• The magnitude of the position vector is distance from the origin, not distance traveled along the path. Those are different ideas.
• Velocity uses the derivatives $x'(t)$ and $y'(t)$, not $x(t)$ and $y(t)$. The position vector tells you location; the velocity vector tells you motion.

Related AP Precalculus Guides

• 4.10 Matrices
• 4.12 Linear Transformations and Matrices
• 4.5 Implicitly Defined Functions
• 4.8 Vectors
• 4.13 Matrices as Functions
• 4.4 Parametrically Defined Circles and Lines