With one input, and vector outputs, we work component-wise.

**component-wise**. Let’s see this in action.

### Limits of vector-valued functions

With a vector-valued function, you have something like

where each component is completely independent of the other components. When computing a limit, we can write:

This is worth stating as a theorem.

We evaluate limits by just taking the limit of each component separately.

Now that we have the notion of limits, we may also define the concept of continuity of vector-valued functions:

Because of the component-wise nature of limits, we can see that a function is continuous if and only if each component function , , is also continuous at .- Constant function
- Identity function
- Power function
- Exponential function
- Logarithmic function
- Sine and cosine
- Both and

In essence, we are saying that the functions listed above are continuous wherever they are defined.

### Derivatives

Recall the limit definition of the derivative: We have a similar limit definition of vector-valued functions: Since limits can be computed component-wise, this derivative can be computed component-wise.

The derivative of a vector-valued function gives a vector that points in the direction that the vector-valued function draws the curve.

Below we see the derivative of the vector-valued function along with an approximation of the limit for small values of :

We also have some (additional) derivative rules:

The derivative of a vector-valued function gives a *tangent vector*. A tangent vector is
a vector that points in the direction that the curve is drawn.

Below we see the derivative of the vector-valued function along with an approximation of the limit for small values of :

- A vector is a
**tangent vector**to the graph of at , if is parallel to . - The
**tangent line**to the graph of at is the line through in a direction parallel to . An equation of the tangent line is - The direction of is given by the
**unit tangent vector**of is given by

We want to be able to predict how a curve drawn by a vector-valued function behaves based on the function’s derivative. However, if the derivative is ever the zero vector, we loose all such information. When this happens we cannot use the tools of calculus. Because of this, we have a special name for functions where the tangent vector is never the zero vector:

Finally, we point out that if a vector-valued function has constant length, then there is a special relationship between the function and its tangent vectors.

and this means that the vectors and are parallelperpendicular .

Now suppose that for all in , and are orthogonal. This means

but we know where the derivative on the left came from! So we may write for some constant . Now we see that is a vector-valued function of constant length on an open interval .

By moving the slider around you can see the vector of constant length that draws the curve and the tangent vector. Note that these vectors are orthogonal.

### Integrals

Since we took the derivative of vector-valued functions by differentiating each component, we will also compute indefinite and definite integrals by computing antiderivatives of each component.

We can also solve initial value problems, check out our next example:

We now leave you with a question: What does integration of a vector-valued function
*mean*? There are many applications, but none as direct as “the area under the curve”
that we used in understanding the integral of a real-valued function. We will explore
this later in our study of calculus.