Vector-Valued Functions - AP Precalculus

1. Identify the point on the curve at $t=2$ for $\mathbf{r}(t)=\langle t-1,2t\rangle$.

   $\langle 1,4\rangle$. Substitute $t=2$: $\langle 2-1, 2(2)\rangle$.

2. What is the cross product of $\textbf{i}$ and $\textbf{j}$?

   $\textbf{k}$. Standard unit vector cross product using right-hand rule.

3. State the formula for the magnitude of $\langle a,b,c\rangle$.

   $\|\langle a,b,c\rangle\|=\sqrt{a^2+b^2+c^2}$. Extends Pythagorean theorem to 3D.

4. State the significance of the Frenet-Serret formulas.

   Describe motion along a curve in space. Provide mathematical framework for analyzing curves in three-dimensional space.

5. What is the result of differentiating a constant vector-valued function?

   Zero vector. Constant functions have zero rate of change.

6. What is the standard component form of a vector-valued function in the plane?

   $\mathbf{r}(t)=\langle x(t),y(t)\rangle$. Each component is a function of parameter $t$.

7. What is the geometric interpretation of the derivative of a vector-valued function?

   The tangent vector to the curve. Shows the direction of motion along the curve.

8. What is the position vector of a particle at time $t$ if its vector-valued function is $\mathbf{r}(t)$?

   $\mathbf{r}(t)$. The function itself gives position at any time.

9. What is the domain and range interpretation of $\mathbf{r}(t)$ for motion?

   Domain: time $t$; range: points $(x(t),y(t),z(t))$. Input is time; output is position in space.

10. State the formula for the derivative of a vector-valued function.

    $\frac{d}{dt} \textbf{r}(t) = \textbf{r}'(t)$. Derivative is taken component-wise for each vector component.

11. Eliminate the parameter: $\mathbf{r}(t)=\langle 2t+1,3t-2\rangle$.

    $3x-2y=7$. From $x=2t+1$, $t=\frac{x-1}{2}$; substitute into $y=3t-2$.

12. What is the speed of a particle with velocity $\mathbf{v}(t)$?

    $\|\mathbf{v}(t)\|$. Speed is the magnitude of velocity vector.

13. What is the torsion of a space curve?

    Measure of how much the curve twists out of the plane of curvature. Quantifies how much a curve deviates from planar motion.

14. What is the unit normal vector $\textbf{N}(t)$?

    Vector orthogonal to unit tangent vector. Points toward the center of curvature of the curve.

15. What is the unit vector in the direction of a nonzero vector $\mathbf{v}$?

    $\frac{\mathbf{v}}{\|\mathbf{v}\|}$. Divide vector by its magnitude to get length 1.

16. State the formula for the magnitude of $\langle a,b\rangle$.

    $\|\langle a,b\rangle\|=\sqrt{a^2+b^2}$. Apply Pythagorean theorem in 2D.

17. What is a vector-valued function?

    A function with a vector output for each input. Each input value maps to a vector instead of a scalar.

18. What is the standard component form of a vector-valued function in space?

    $\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle$. Extends 2D form by adding a third component function.

19. What is the elimination-of-parameter goal for $\mathbf{r}(t)=\langle x(t),y(t)\rangle$?

    A Cartesian relation $F(x,y)=0$ with no $t$. Eliminate $t$ to get direct $x$-$y$ relationship.

20. What is the curvature $\text{k}(t)$ of a vector-valued function?

    Measure of how a curve deviates from being a straight line. Higher curvature means the curve bends more sharply.

21. Find the unit vector in the direction of $\langle 6,8\rangle$.

    $\left\langle \frac{3}{5},\frac{4}{5}\right\rangle$. $\frac{\langle 6,8\rangle}{10} = \langle 0.6, 0.8\rangle$.

22. Define a smooth vector-valued function.

    A function with continuous derivatives. All component functions must be differentiable and continuous.

23. What operation combines two vector-valued functions by addition?

    Component-wise addition. Add corresponding components of each vector function.

24. Identify the parametric equations for $\mathbf{r}(t)=\langle x(t),y(t)\rangle$.

    $x=x(t),\ y=y(t)$. Components become parametric equations.

25. What does the magnitude of a vector-valued function represent?

    The length of the vector at each point. Represents the distance from the origin to the vector tip.

26. What is the geometric significance of the cross product $\textbf{r}(t) \times \textbf{s}(t)$?

    Gives a vector perpendicular to both $\textbf{r}$ and $\textbf{s}$. Result vector is orthogonal to both input vectors.

27. What is the normal vector for a two-dimensional curve given by $\textbf{r}(t)$?

    Perpendicular to the tangent vector. Normal vector is orthogonal to the direction of motion.

28. Find the speed at time $t$ if $\mathbf{v}(t)=\langle 3,4\rangle$.

    $5$. $\|\langle 3,4\rangle\| = \sqrt{9+16} = 5$.

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