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Section11.1Vector-Valued Functions

We are very familiar with real valued functions, that is, functions whose output is a real number. This section introduces vector-valued functions — functions whose output is a vector.

Definition11.1.1Vector-Valued Functions

A vector-valued function is a function of the form \begin{equation*} \vec r(t) = \la\, f(t),g(t)\,\ra \text{ or } \vec r(t) = \la \,f(t),g(t),h(t)\,\ra, \end{equation*} where \(f\text{,}\) \(g\) and \(h\) are real valued functions.

The domain of \(\vec r\) is the set of all values of \(t\) for which \(\vec r(t)\) is defined. The range of \(\vec r\) is the set of all possible output vectors \(\vec r(t)\text{.}\)

Evaluating and Graphing Vector-Valued Functions

Evaluating a vector-valued function at a specific value of \(t\) is straightforward; simply evaluate each component function at that value of \(t\text{.}\) For instance, if \(\vec r(t) = \la t^2,t^2+t-1\ra\text{,}\) then \(\vec r(-2) = \la 4,1\ra\text{.}\) We can sketch this vector, as is done in Figure 11.1.2(a). Plotting lots of vectors is cumbersome, though, so generally we do not sketch the whole vector but just the terminal point. The graph of a vector-valued function is the set of all terminal points of \(\vec r(t)\text{,}\) where the initial point of each vector is always the origin. In Figure 11.1.2(b) we sketch the graph of \(\vec r\) ; we can indicate individual points on the graph with their respective vector, as shown.

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Figure11.1.2Sketching the graph of a vector-valued function.

Vector–valued functions are closely related to parametric equations of graphs. While in both methods we plot points \(\big(x(t), y(t)\big)\) or \(\big(x(t),y(t),z(t)\big)\) to produce a graph, in the context of vector-valued functions each such point represents a vector. The implications of this will be more fully realized in the next section as we apply calculus ideas to these functions.

Example11.1.5Graphing vector-valued functions

Graph \(\ds \vec r(t) = \la t^3-t, \frac{1}{t^2+1}\ra\text{,}\) for \(-2\leq t\leq 2\text{.}\) Sketch \(\vec r(-1)\) and \(\vec r(2)\text{.}\)

Example11.1.7Graphing vector-valued functions.

Graph \(\vec r(t) = \la \cos(t) ,\sin(t) ,t\ra\) for \(0\leq t\leq 4\pi\text{.}\)


Subsection11.1.1Algebra of Vector-Valued Functions

Definition11.1.9Operations on Vector-Valued Functions

Let \(\vec r_1(t)=\la f_1(t),g_1(t)\ra\) and \(\vec r_2(t)=\la f_2(t),g_2(t)\ra\) be vector-valued functions in \(\mathbb{R}^2\) and let \(c\) be a scalar. Then:

  1. \(\vec r_1(t) \pm \vec r_2(t) = \la\, f_1(t)\pm f_2(t),g_1(t)\pm g_2(t)\,\ra\text{.}\)

  2. \(c\vec r_1(t) = \la\, cf_1(t),cg_1(t)\,\ra\text{.}\)

A similar definition holds for vector-valued functions in \(\mathbb{R}^3\text{.}\)

This definition states that we add, subtract and scale vector-valued functions component–wise. Combining vector-valued functions in this way can be very useful (as well as create interesting graphs).

Example11.1.10Adding and scaling vector-valued functions.

Let \(\vec r_1(t) = \la\,0.2t,0.3t\,\ra\text{,}\) \(\vec r_2(t) = \la\,\cos(t) ,\sin(t) \,\ra\) and \(\vec r(t) = \vec r_1(t)+\vec r_2(t)\text{.}\) Graph \(\vec r_1(t)\text{,}\) \(\vec r_2(t)\text{,}\) \(\vec r(t)\) and \(5\vec r(t)\) on \(-10\leq t\leq10\text{.}\)

Example11.1.15Adding and scaling vector-valued functions.

A cycloid is a graph traced by a point \(p\) on a rolling circle, as shown in <<Unresolved xref, reference "fig_vvf4"; check spelling or use "provisional" attribute>>. Find an equation describing the cycloid, where the circle has radius 1.

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\captionof{figure}{Tracing a cycloid.}



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Figure11.1.16The cycloid in Example 11.1.15.

A vector-valued function \(\vec r(t)\) is often used to describe the position of a moving object at time \(t\text{.}\) At \(t=t_0\text{,}\) the object is at \(\vec r(t_0)\text{;}\) at \(t=t_1\text{,}\) the object is at \(\vec r(t_1)\text{.}\) Knowing the locations \(\vec r(t_0)\) and \(\vec r(t_1)\) give no indication of the path taken between them, but often we only care about the difference of the locations, \(\vec r(t_1)-\vec r(t_0)\text{,}\) the displacement.


Let \(\vec r(t)\) be a vector-valued function and let \(t_0\lt t_1\) be values in the domain. The displacement \(\vec d\) of \(\vec r\text{,}\) from \(t=t_0\) to \(t=t_1\text{,}\) is \begin{equation*} \vec d=\vec r(t_1)-\vec r(t_0). \end{equation*}

When the displacement vector is drawn with initial point at \(\vec r(t_0)\text{,}\) its terminal point is \(\vec r(t_1)\text{.}\) We think of it as the vector which points from a starting position to an ending position.

Example11.1.18Finding and graphing displacement vectors

Let \(\vec r(t) = \la \cos(\frac{\pi}{2}t),\sin(\frac{\pi}2 t)\ra\text{.}\) Graph \(\vec r(t)\) on \(-1\leq t\leq 1\text{,}\) and find the displacement of \(\vec r(t)\) on this interval.


Measuring displacement makes us contemplate related, yet very different, concepts. Considering the semi–circular path the object in Example 11.1.18 took, we can quickly verify that the object ended up a distance of 2 units from its initial location. That is, we can compute \(\vnorm{d} = 2\text{.}\) However, measuring distance from the starting point is different from measuring distance traveled. Being a semi–circle, we can measure the distance traveled by this object as \(\pi\approx 3.14\) units. Knowing distance from the starting point allows us to compute average rate of change.

Definition11.1.20Average Rate of Change

Let \(\vec r(t)\) be a vector-valued function, where each of its component functions is continuous on its domain, and let \(t_0\lt t_1\text{.}\) The average rate of change of \(\vec r(t)\) on \([t_0,t_1]\) is \begin{equation*} \text{ average rate of change } = \frac{\vec r(t_1) - \vec r(t_0)}{t_1-t_0}. \end{equation*}

Example11.1.21Average rate of change

Let \(\vec r(t) = \la \cos(\frac{\pi}2t),\sin(\frac{\pi}2t)\ra\) as in Example 11.1.18. Find the average rate of change of \(\vec r(t)\) on \([-1,1]\) and on \([-1,5]\text{.}\)


We considered average rates of change in Sections 1.1 and 2.1 as we studied limits and derivatives. The same is true here; in the following section we apply calculus concepts to vector-valued functions as we find limits, derivatives, and integrals. Understanding the average rate of change will give us an understanding of the derivative; displacement gives us one application of integration.


In the following exercises, sketch the vector-valued function on the given interval.

In the following exercises, sketch the vector-valued function on the given interval in \(\mathbb{R}^3\text{.}\) Technology may be useful in creating the sketch.

In the following exercises, find \(\norm{\vec r(t)}\text{.}\)

In the following exercises, create a vector-valued function whose graph matches the given description.

In the following exercises, find the average rate of change of \(\vec r(t)\) on the given interval.