9.6 Analyzing Motion of Vector-Valued Functions

• A vector-valued function in two dimensions is expressed as \langle x(t), y(t) \rangle
• What do vector-valued functions map real numbers to?

  Vectors in plane or space
• What are $x(t)$, y(t)</latex>, and $z(t)$ called in a vector-valued function?


  Component functions
• The vector-valued function $\vec{r}(t) =$$\langle t, t^{2}, t^{3} \rangle$ represents a spiral curve
• Vector-valued functions map real numbers to vectors in the plane or space.


  True
• What do vector-valued functions represent when used in motion analysis?

  Motion of an object
• What path does the vector-valued function $\vec{r}(t) =$$\langle 2t, t^{2} \rangle$ represent in the plane?


  Parabola
• What path in space does the vector-valued function \vec{r}(t) = \langle t, t^{2}, t^{3} \rangle</latex> represent?

  Spiral curve
• The vector-valued function \vec{r}(t) = \langle 2t, t^{2} \rangle</latex> represents a parabola
• The vector-valued function $\vec{r}_{2}(t) =$$\langle \cos(t), \sin(t) \rangle$ represents a circle
• In a vector-valued function, the variables $x(t)$, $y(t)$, and $z(t)$ are called the component functions
• Match the vector-valued function with its corresponding path:

  $\vec{r}_{1}(t) =$$\langle t, t^{2} \rangle$ ↔️ Parabola

  $\vec{r}_{2}(t) =$$\langle \cos(t), \sin(t) \rangle$ ↔️ Circle
• What type of path does the vector-valued function $\vec{r}(t) =$$\langle t, t^{2}, t^{3} \rangle$ represent?


  Spiral curve
• The velocity vector of an object is found by differentiating its position vector with respect to time.


  True
• The velocity vector's derivative gives the instantaneous direction of motion and the object's speed.
• What is the speed of an object with velocity vector $\vec{v}(t) =$$\langle 6t^{2}, 5 \rangle$?


  \sqrt{(6t^{2})^{2} + 5^{2}}
• The variables $x(t)$, $y(t)$, and z(t)</latex> in a vector-valued function are called the component functions.
• In two dimensions, a vector-valued function is expressed as \vec{r}(t) = \langle x(t), y(t) \rangle</latex>


  True
• Match the vector-valued function with its corresponding path in the plane:

  $\vec{r}(t) =$$\langle 2t, t^{2} \rangle$ ↔️ Parabola

  $\vec{r}(t) =$$\langle \cos(t), \sin(t) \rangle$ ↔️ Circle
• What path in space does the vector-valued function \vec{r}(t) = \langle \cos(t), \sin(t), t \rangle</latex> represent?

  Helix
• The component functions of a vector-valued function define the path
• Match the vector-valued function with its corresponding path in the plane:

  $\vec{r}(t) =$$\langle 2t, t^{2} \rangle$ ↔️ Parabola

  $\vec{r}(t) =$$\langle \cos(t), \sin(t) \rangle$ ↔️ Circle
• Vector-valued functions are used to represent the motion of an object in space.


  True
• What is the path of the vector-valued function $\vec{r}_{1}(t) =$$\langle t, t^{2} \rangle$?


  Parabola
• What is the general three-dimensional form of a vector-valued function?

  $\vec{r}(t) =$$\langle x(t), y(t), z(t) \rangle$
• Vector-valued functions map real numbers to vectors in the plane or space.


  True
• Vector-valued functions are used to represent the motion of an object.
• What is the three-dimensional form of a vector-valued function used to represent motion in space?

  $\vec{r}(t) =$$\langle x(t), y(t), z(t) \rangle$
• What is the velocity vector $\vec{v}(t)$ for the vector-valued function $\vec{r}(t) =$$\langle 2t^{3}, 5t \rangle$?


  $\vec{v}(t) =$$\langle 6t^{2}, 5 \rangle$
• Speed is defined as the magnitude of the velocity vector.


  True
• What does a vector-valued function map real numbers to?

  Vectors in space
• Vector-valued functions are used to represent the motion of an object in the plane or in space.
• The component functions of a vector-valued function define the path of the object.
• The velocity vector is found by differentiating the position vector with respect to time.

  True
• Differentiating a component-wise in a vector-valued function means differentiating each component separately with respect to t.
• Speed is the magnitude of the velocity vector.
• Speed is the magnitude of the velocity
• Match the vector-valued function with its speed:

  \(\langle 2t^{3}, 5t \rangle\) ↔️ \(\sqrt{(6t^{2})^{2} + 5^{2}}\)

  \(\langle \cos(t), \sin(t) \rangle\) ↔️ 1
• The acceleration vector $\vec{a}(t)$ is equal to the second derivative of the position function components, which is \langle x''(t), y''(t) \rangle</latex>.second
• The speed of an object is the magnitude of its velocity