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.