Vector Valued Function

Vectors

Fig. 1. A vector can be thought of like an arrow, pointing from one place to another.

Vectors can be written in two different ways,

• Column Vector Form: \( \begin{bmatrix} x \\ y \end{bmatrix}, \)

• Component Form: \( x \vec{i} + y \vec{j}. \)

These two vectors are equivalent. Numerically, vectors can be added and subtracted by adding or subtracting the individual components. Similarly, they can be multiplied by scalar quantities by multiplying the individual components. In component form, this looks just like collecting like terms and expanding brackets.

Graphically, adding vectors is done by stacking them tip to tip, and subtracting by stacking them tip to tip, but pointing the second vector in the opposite direction. Multiplying numbers by a scalar \(\lambda\) is the same as stacking \(\lambda\) of the same vectors, tip to tip, and if \(\lambda\) is negative, the product will be pointing in the opposite direction.

Finally, given a vector \( v = x \vec{i} + y \vec{j},\) the magnitude \(|\vec{v}|\) and direction angle \(\theta\) of a vector can be calculated using the following formulas:

\[ \begin{align} | \vec{v} | & = \sqrt{ x^2 + y^2 }, \\ \theta & = \tan^{-1}\left({\frac{x}{y}}\right) \end{align} \]

For more information on all of this, see Vectors.

What are Vector-Valued Functions?

Vector-valued functions are just like real-valued functions, but output a vector instead of a scalar.

Given this definition, you can deduce the domain and range of a vector-valued function.

• The domain of a vector-valued function is a subset of \(\mathbb{R},\)

• The range of an \(n\)-dimensional vector-valued function is a subset of \(\mathbb{R}^n.\)

Here you will focus on vectors in 2 dimensions, meaning the range of the functions will be a subset of \(\mathbb{R}^2.\) It is important to note that it is a subset of \(\mathbb{R}^2\) and not the whole of \(\mathbb{R}^2,\) since you will encounter many vector-valued functions that cannot output to every point in \(\mathbb{R}^2.\)

Examples of Vector-Valued Functions

There are many different types of vector-valued functions, but here you will look at some of the simplest.

Straight Lines

The vector-valued formula for a straight line is

\[ \vec{r}(t) = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} + t \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}. \]

Here, \(\vec{a} = a_1 \vec{i} + a_2 \vec{b} \) is the position vector of a point \(a\) on the line, and \( \vec{b} = b_1 \vec{i} + b_2 \vec{j} \) is a vector that is parallel to the line.

Fig. 2. A line is defined by a vector-valued function using a point on the line, \(a,\) and a vector parallel to the line, \(\vec{b}.\)

Circles and Ellipses

The vector-valued equation for a circle with radius \(a\) is

\[ \vec{r}(t) = \begin{bmatrix} a \cos{t} \\ a \sin{t} \end{bmatrix} \]

Fig. 3. The vector-valued function for a circle can be made using the sine and cosine functions.

An ellipse can be defined similarly, but using \(a\) as the intercept on the \(x\)-axis and \(b\) as the intercept on the \(y\)-axis.

\[ \vec{r}(t) = \begin{bmatrix} a \cos{t} \\ b \sin{t} \end{bmatrix} \]

Fig. 4. The vector-valued function for an ellipse can be defined similarly to that of a circle, but taking into account the different axis intercepts.

Spirals

There are many ways to define spirals in mathematics, but an easy way is to define them similarly to spirals and circles, but with a \(t\) term in front of the trigonometric functions.

\[ \vec{r}(t) = \begin{bmatrix} a t \cos{t} \\ b t \sin{t} \end{bmatrix} \]

Fig. 5, The graph of an spiral, where \(a = b = \frac{1}{2}. \)

Graphing Vector-Valued Functions

When you first learnt to graph Cartesian equations such as \(y = f(x),\) you likely started by drawing a table of values for \(x,\) and then filling in the corresponding values of \(y.\) You could then plot these points and join them up, to create an estimation of the curve. You can do the exact same thing to graph vector-valued functions, but instead starting with the variable \(t\) and using these values of \(t\) to calculate the corresponding values of \(x\) and \(y.\) Let's look at an example of this.

To see more examples, see Graphing Vector-Valued Functions.

Vector-Valued Functions Formula

The most important formula for vector-valued functions is the formula for arc length, or the length of a curve between two points.

The length of the curve between the points \(t=a\) and \(t=b.\)

The length \(L\) of a curve \(\vec{r}(t) = f(t) \vec{i} + g(t) \vec{j} \) between two point \(a\) and \(b\) is

\[ L = \int_a^b \sqrt{[f'(t)]^2 + [g'(t)]^2} \, \mathrm{d}t. \]

This measures the whole length of the curve as if you had laid a piece of string on the curve and then cut it off and measured it. Let's look at some examples using this formula.

Derivatives of Vector-Valued Function

The derivative of vector-valued functions can be found by differentiating each component of the vector-valued function. The derivative of \( \vec{r}(t) = f(t) \vec{i} + g(t) \vec{j} \) is:

\[ \frac{\mathrm{d}\vec{r}}{\mathrm{d}t}(t) = \frac{\mathrm{d}f}{\mathrm{d}t}(t) \vec{i} + \frac{\mathrm{d}g}{\mathrm{d}\mathrm{d}}(t) \vec{j}, \]

assuming that the derivatives of \(f(t)\) and \(g(t)\) with respect to \(t\) exist. This makes sense logically, as it is just like using the addition rule when differentiating any other function. The derivative of a vector-valued function at a point will point in the direction of travel of the function, at a tangent to the curve.

If the vector valued function, call it \(\vec{s}(t),\) represents position on the \(xy\) plane at time \(t,\) then the derivative of this function will be the velocity vector \(\vec{v}(t).\) The magnitude of the velocity vector at time \(t\) is the speed of travel at time \(t.\) Similarly, the differential of the velocity vector will be the acceleration vector, \( \vec{a}(t). \) Let's take a look at differentiating some vector-valued functions.

To learn more about differentiating vector-valued functions, see Calculus of Vector-Valued Functions.