A vector-valued function (or vector function) is a function that takes one or more variables and returns a vector. Here is a common example of a vector-valued function in three dimensions:

Example:

Let $\vec{r}(t)$ be a vector-valued function defined by: $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle = \langle t, t^2, t^3 \rangle$ where:

• $f(t) = t$
• $g(t) = t^2$
• $h(t) = t^3$

In this case: $\vec{r}(t) = \langle t, t^2, t^3 \rangle$

Explanation of Components:

• Input: The function $\vec{r}(t)$ is parameterized by $t$, which usually represents time or another variable.
• Output: For each value of $t$, $\vec{r}(t)$ gives a vector with three components corresponding to the $x$, $y$, and $z$ coordinates in 3D space.

Interpretation:

This particular vector function describes a curve in 3D space. The path traced out by $\vec{r}(t)$ as $t$ varies gives the shape of the curve. In this case, the curve is a space curve whose projection on the $xy$-plane is a parabola, and whose $z$-coordinate is increasing with $t$ cubed.

Calculus with Vector-Valued Functions:

1. Derivative: The derivative of $\vec{r}(t)$ with respect to $t$, denoted $\vec{r}'(t)$, is calculated by differentiating each component: $\vec{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle = \langle 1, 2t, 3t^2 \rangle$
2. Integral: The integral of $\vec{r}(t)$ with respect to $t$ is obtained by integrating each component separately.

Would you like more details on any specific aspect of this function or vector functions in general?

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Here are some related questions for further exploration:

1. How do you calculate the magnitude of a vector-valued function?
2. What is the interpretation of the derivative of a vector-valued function?
3. How can you find the unit tangent vector to a vector function at a given point?
4. What role do vector-valued functions play in physics, especially in representing motion?
5. How do you compute the arc length of a curve given by a vector function?

Tip: For curves in 3D, vector-valued functions provide a compact and powerful way to represent trajectories and analyze their properties with calculus.