A **vector-valued function** is a function that takes one or more variables as input and returns a vector as the output, rather than a scalar. The components of the vector output are typically functions of the input variable(s). Vector-valued functions are commonly used to describe physical quantities like position, velocity, and acceleration in space.

### General Form:
A vector-valued function is usually written as:
\[
\mathbf{r}(t) = \langle f_1(t), f_2(t), \dots, f_n(t) \rangle
\]
where:
- \(t\) is the input variable (often time in physical contexts).
- \(f_1(t), f_2(t), \dots, f_n(t)\) are scalar functions representing the components of the vector.

For example, a **3-dimensional** vector-valued function can be written as:
\[
\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle
\]
where \(x(t)\), \(y(t)\), and \(z(t)\) are scalar functions of the input variable \(t\).

### Example:
In 2D, a vector-valued function might be:
\[
\mathbf{r}(t) = \langle \cos(t), \sin(t) \rangle
\]
This describes a circular path, where the \(x\)-component is \(\cos(t)\) and the \(y\)-component is \(\sin(t)\). As \(t\) varies, the vector traces out a circle.

### Applications:
1. **Physics**: Describing motion, where the vector function represents the position of an object at time \(t\). The derivative of this function can give velocity and acceleration vectors.
2. **Geometry**: Used to represent curves in 2D, 3D, or higher dimensions, where the vector-valued function defines the position of points on the curve.

### Derivatives and Integrals:
- The derivative of a vector-valued function is computed by differentiating each component of the vector separately. This gives the rate of change of the vector with respect to the input variable.
- Similarly, integrals of vector-valued functions are computed by integrating each component individually.

### Key Concepts:
- **Domain**: The set of input values (e.g., \(t\)) for which the function is defined.
- **Range**: The set of all output vectors the function can produce.
  
Would you like more details or further clarification on any part?

### Follow-Up Questions:
1. What is the derivative of a vector-valued function, and how is it interpreted?
2. How are vector-valued functions used to describe motion in physics?
3. Can you explain how vector-valued functions describe curves in space?
4. What is the significance of the integral of a vector-valued function?
5. How does a vector-valued function differ from a scalar-valued function?

### Tip:
When dealing with vector-valued functions, visualizing their components individually can help better understand the overall behavior of the function.

Explore vector-valued functions, including their general form and applications in physics and geometry. Learn how to compute derivatives and integrals of vector-valued functions and how they describe motion and curves in space. Suitable for high school and undergraduate students.

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