Dated: 30-10-2024

Vector Valued Functions

The functions¹ which have real numbers as their domain¹ and 2D or 3D vectors² as their range,¹ are called vector valued functions.

2D

\[\vec{r}(t) = \langle\vec{x}(t), \vec{y}(t)\rangle = x(t)\hat{i} + y(t)\hat{j}\]

3D

\[\vec{r}(t) = \langle\vec{x}(t), \vec{y}(t), \vec{z}(t)\rangle = x(t)\hat{i} + y(t)\hat{j} + z\hat{k}\]

Graphs of Vector Valued Functions

Let us say we have this vector valued function.

\[\vec{r}(t) = \langle\vec{x}(t), \vec{y}(t)\rangle = x(t)\hat{i} + y(t)\hat{j}\]

As \(t\) varies, we get a vector² whose tail is at origin and head moves along a curve \(C\).
This vector is called the position or radius vector \(\vec{OC}\) and \(C\) is the graph.

Example

\[\vec{r}(t) = \hat{i} \cos(t) + \hat j \sin (t)\]

\[\because \vec{r}(t) = x \hat i + y \hat j\]

\[x \hat i + y \hat j= \hat{i} \cos(t) + \hat j \sin (t)\]

This shows

\[x = \cos (t)\]

\[y = \sin (t)\]

which are parametric equations of a circle where \(t\) is \(\theta\).

Example

\[\vec{r}(t) = \hat{i} \cdot a \cdot \cos(t) + \hat j \cdot b \cdot \sin (t) + \hat z \cdot c \cdot t\]

Where \(a, b, c\) are constants.
Let us inspect this equation closely, the \(\hat i\) and \(\hat j\) components create a circle.
Then we have \(\hat z\) whose coefficient changes as \(t\) changes, defining a changing height.
The graph it creates is a circular helix.

References

Read more about notations and symbols.

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1. Read more about functions. ↩↩↩

2. Read more about vectors. ↩↩