Unit 8 · Topic 4.9 · Optional
Vector-Valued Functions
Use a single input to produce horizontal and vertical components, then interpret position, displacement, velocity, direction, distance from the origin, and speed in planar motion.
Learning Goals
- Represent a vector-valued function in component and unit-vector forms.
- Evaluate position vectors and connect them to parametric equations.
- Distinguish position, displacement, distance from the origin, and distance traveled.
- Interpret horizontal and vertical velocity components from their signs.
- Calculate speed as the magnitude of a velocity vector.
- Use tables, formulas, and curves together to justify conclusions about planar motion.
1. A Function Whose Output Is a Vector
A real-valued function returns one number. A vector-valued function in the plane returns an ordered pair:
The scalar functions \(x(t)\) and \(y(t)\) are the component functions. The domain of r is the intersection of their domains because both coordinates must exist at the same input t.
2. Read the Input and Output Correctly
| Symbol | Role | Possible units |
|---|---|---|
| t | Shared input or parameter | seconds |
| \(x(t)\) | Horizontal component | meters |
| \(y(t)\) | Vertical component | meters |
| \(\mathbf r(t)\) | Vector output | ordered position in meters |
The parameter is not automatically an x-coordinate. It becomes the x-coordinate only when \(x(t)=t\).
3. Position Vector and Parametric Form
If \(\mathbf p(t)=\langle x(t),y(t)\rangle\) gives position, then the moving point at time t is \((x(t),y(t))\). Thus,
The vector is drawn from the origin to the current point; the point's path is the associated parametric curve.
4. One Formula Encodes a Path and Its Timing
The geometric curve shows where the object travels. Increasing t orders the points and reveals when and in which direction the path is traversed. Two vector-valued functions can trace the same curve with different timing or direction.
5. Evaluate a Position Function
Let
At \(t=3\),
The object is at the point \((6,9)\). Eliminating t gives \(t=x/2\), so the path satisfies \(y=x^2/4\) with \(x\ge0\), but the vector form also preserves the time information.
6. Distance from the Origin
The magnitude of a position vector gives the object's straight-line distance from the origin:
For \(\mathbf p(3)=\langle6,9\rangle\), the distance is \(\sqrt{117}=3\sqrt{13}\), not \(6+9\).
7. Displacement Is a Vector
From time \(t=a\) to \(t=b\), displacement is final position minus initial position:
For \(\mathbf p(t)=\langle2t,t^2\rangle\), displacement from \(t=1\) to \(t=3\) is \(\langle6,9\rangle-\langle2,1\rangle=\langle4,8\rangle\).
8. Displacement, Its Magnitude, and Distance Traveled
| Quantity | Meaning | Type |
|---|---|---|
| \(\mathbf p(b)-\mathbf p(a)\) | Net change in position | Vector |
| \(\|\mathbf p(b)-\mathbf p(a)\|\) | Straight-line distance between endpoints | Scalar |
| Distance traveled | Length of the actual route | Scalar |
Distance traveled is at least the magnitude of displacement. They are equal only when the route from start to finish is straight and never reverses.
9. Average Velocity over an Interval
The average velocity vector records average horizontal and vertical change per unit time:
Using the displacement \(\langle4,8\rangle\) from \(t=1\) to \(t=3\), \(\mathbf v_{\mathrm{avg}}=\langle2,4\rangle\) units per time unit.
10. Interpret a Velocity Vector
A velocity model can be written
| Component sign | Motion |
|---|---|
| \(v_x(t)>0\) | Right |
| \(v_x(t)<0\) | Left |
| \(v_y(t)>0\) | Up |
| \(v_y(t)<0\) | Down |
A zero component means no motion in that coordinate at that instant; it does not necessarily mean the object is stopped.
11. Speed Is the Magnitude of Velocity
Speed is a nonnegative scalar. Velocity includes direction, so two objects can have the same speed but different velocity vectors.
12. Worked Velocity Example
Suppose
At \(t=1\), \(\mathbf v(1)=\langle2,-3\rangle\), so the object moves right and down with speed \(\sqrt{13}\). At \(t=2\), \(\mathbf v(2)=\langle0,0\rangle\), so it is momentarily stopped. At \(t=3\), \(\mathbf v(3)=\langle-2,3\rangle\), so it moves left and up, again with speed \(\sqrt{13}\).
13. Circular Position and Tangential Velocity
Consider the position and compatible velocity models
The position magnitude is always 3, so the object remains on a radius-3 circle. Also,
so the velocity is perpendicular to the radius and tangent to the circular path. Its speed is \(\sqrt{9\sin^2t+9\cos^2t}=3\).
14. Context Example: A Drone
A drone's position in meters is
and its constant velocity is \(\mathbf v(t)=\langle6,8\rangle\) meters per second. It moves right and up at speed 10 meters per second. At \(t=5\),
so the drone is 100 meters from the origin. That value is not the same question as how far it has traveled since \(t=0\).
15. AP Workflow and Common Errors
- Identify whether the function represents position or velocity.
- Evaluate every component at the same input.
- Use position magnitude only for distance from the origin.
- Subtract final minus initial position for displacement.
- Divide displacement by elapsed time for average velocity.
- Use velocity-component signs to identify horizontal and vertical direction.
- Take velocity magnitude for speed and attach appropriate units.
- Do not treat t as the x-coordinate unless \(x(t)=t\).
- Do not add components when finding a vector's magnitude.
- Do not confuse position with distance from the origin.
- Do not confuse displacement magnitude with distance traveled.
- Do not call one zero velocity component a complete stop.
- Do not report speed as a signed quantity or a vector.
Key Takeaways
- Use vector-valued functions to describe changing position, direction, and other multicomponent quantities.
- Core relationship: \(\mathbf r(t)=\langle f(t),g(t)\rangle\)
- Error check: Interpret each component with its own units and meaning.
- For \(\mathbf p(t)=\langle t^2-1,3t\rangle\), find the position and distance from the origin at \(t=2\).
- Find the displacement and average velocity from \(t=1\) to \(t=4\).
- Eliminate the parameter from \(\mathbf p(t)=\langle2t,t^2\rangle\) and state the path restriction when \(t\ge0\).
- If \(\mathbf v(5)=\langle-7,0\rangle\), describe the direction of motion and calculate the speed.
- For \(\mathbf v(t)=\langle t-2,4-t\rangle\), identify the direction of motion at \(t=1,3,\) and 5, and determine whether the object is ever stopped.