Writing Curves in Component Form

Introduction

A space curve can be described by a moving position vector, or by three coordinate rules that say how its \(x\)-, \(y\)-, and \(z\)-coordinates depend on a parameter \(t\). These descriptions contain the same information when their components are matched in order.

For example, the vector rule \(\mathbf{r}(t)=\langle t, t^2, 3-t\rangle\) says that at parameter value \(t\), the position has first coordinate \(t\), second coordinate \(t^2\), and third coordinate \(3-t\). In component form, those rules are \(x=x(t)\), \(y=y(t)\), and \(z=z(t)\).

This tutorial focuses on translating between these forms and constructing one form from the other. The key is to keep each coordinate paired with the correct component; a change in order changes the vector rule.

Coordinate Functions and Vector Notation

A vector-valued function assigns a vector to each allowed parameter value. For a curve in three-dimensional space, the vector has three coordinates. We use \(\mathbf{r}(t)\) for the position vector and \(x(t)\), \(y(t)\), and \(z(t)\) for its coordinate functions.

Definition: A vector-valued function describing a space curve can be written in component form as

\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle.

Here \(x(t)\), \(y(t)\), and \(z(t)\) are the first, second, and third coordinates of \(\mathbf{r}(t)\), respectively. Equivalently, the coordinate equations are \(x=x(t)\), \(y=y(t)\), and \(z=z(t)\).

The angle brackets list the components in order. The first entry is the \(x\)-coordinate, the second is the \(y\)-coordinate, and the third is the \(z\)-coordinate. The same vector can also be written using the standard coordinate unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\), or as a column vector:

\mathbf{r}(t) = x(t)\mathbf{i}+y(t)\mathbf{j}+z(t)\mathbf{k} = \begin{bmatrix} x(t)\\ y(t)\\ z(t) \end{bmatrix}.

These are three notations for the same vector. The angle-bracket form is often the quickest for reading off coordinates; the unit-vector form emphasizes how the vector is built from coordinate directions; and the column form is useful in matrix calculations.

The coordinate equations are not three separate curves with unrelated parameters. They describe the coordinates of one vector, all evaluated at the same parameter value \(t\). For instance, if \(t=2\), then each coordinate function is evaluated at \(2\) to produce the three entries of one position vector.

Reading a Vector Rule in Component Form

To translate from vector notation to coordinate equations, identify the first, second, and third entries. Then write each entry as its corresponding coordinate function. If the vector is expressed using \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\), read off the coefficient of each unit vector. If it is a column vector, read the entries from top to bottom.

Worked Example: Read the Components of a Vector Rule

Write the coordinate equations for \(\mathbf{r}(t)=\langle 2t-1,\ t^2+4,\ \sin t\rangle\).

The first entry is \(2t-1\), so \(x(t)=2t-1\), or \(x=2t-1\). The second entry is \(t^2+4\), so \(y(t)=t^2+4\), or \(y=t^2+4\). The third entry is \(\sin t\), so \(z(t)=\sin t\), or \(z=\sin t\).

\begin{aligned} x&=2t-1,\\ y&=t^2+4,\\ z&=\sin t. \end{aligned}

To check the translation, put the three coordinate rules back in their original order: \(\langle x(t),y(t),z(t)\rangle=\langle 2t-1,t^2+4,\sin t\rangle\). This is exactly the given vector rule, so the components match.

The same reading process works when the rule uses unit vectors or a column. The form may look different, but the order of coordinates does not change.

Worked Example: Read Unit-Vector and Column Forms

For each expression below, write \(x=x(t)\), \(y=y(t)\), and \(z=z(t)\).

First consider \(\mathbf{r}(t)=(t+3)\mathbf{i}-2t\mathbf{j}+(t^2-1)\mathbf{k}\). The coefficient of \(\mathbf{i}\) gives the first coordinate, the coefficient of \(\mathbf{j}\) gives the second, and the coefficient of \(\mathbf{k}\) gives the third. Thus \(x=t+3\), \(y=-2t\), and \(z=t^2-1\).

Now consider the column form \(\mathbf{r}(t)=\begin{bmatrix}4-t\\ 2t+1\\ t^3\end{bmatrix}\). The top entry gives \(x=4-t\), the middle entry gives \(y=2t+1\), and the bottom entry gives \(z=t^3\).

For the first expression, the component vector is \(\langle t+3,-2t,t^2-1\rangle\); its unit-vector expansion reproduces the given expression. For the second, the column entries in order are \(\langle 4-t,2t+1,t^3\rangle\); placing them into a column reproduces the given column. Both checks confirm the coordinate assignments.

Constructing a Vector Rule from Coordinate Equations

To go in the other direction, place the right-hand sides of \(x=x(t)\), \(y=y(t)\), and \(z=z(t)\) into an ordered triple. The first rule goes first, the second goes second, and the third goes third. The result is a vector-valued function of the same parameter.

Rule: If the coordinate equations are \(x=f(t)\), \(y=g(t)\), and \(z=h(t)\), then the corresponding vector-valued function is

\mathbf{r}(t)=\langle f(t),g(t),h(t)\rangle.

The three coordinate rules must use the same parameter \(t\), and the entries must remain in \(x\)-, \(y\)-, \(z\)-order.

Worked Example: Assemble Three Coordinate Rules

Construct \(\mathbf{r}(t)\) from \(x=t^2-2\), \(y=5t\), and \(z=1-t^3\).

The \(x\)-rule is \(t^2-2\), so this is the first entry. The \(y\)-rule is \(5t\), so this is the second entry. The \(z\)-rule is \(1-t^3\), so this is the third entry. Therefore,

\mathbf{r}(t)=\langle t^2-2,\ 5t,\ 1-t^3\rangle.

In unit-vector notation, the same vector is \(\mathbf{r}(t)=(t^2-2)\mathbf{i}+5t\mathbf{j}+(1-t^3)\mathbf{k}\). In column notation, it is \(\mathbf{r}(t)=\begin{bmatrix}t^2-2\\5t\\1-t^3\end{bmatrix}\).

To check, read the first entry back as \(x=t^2-2\), the second as \(y=5t\), and the third as \(z=1-t^3\). These are precisely the three given coordinate rules, in their original order.

It is often useful to verify a constructed rule at a particular parameter value. This does not replace checking the symbolic components, but it can catch a misplaced entry. For example, at \(t=2\) in the preceding example, \(x=2^2-2=2\), \(y=5(2)=10\), and \(z=1-2^3=-7\). Thus the vector at that value is \(\mathbf{r}(2)=\langle 2,10,-7\rangle\). Substituting \(t=2\) into the assembled vector gives the same entries.

Why the Translation Works

The reason these conversions are valid is that a vector in three-dimensional coordinate space is determined uniquely by its three coordinates. The standard unit vectors point in the three coordinate directions, so multiplying them by the coordinate values and adding gives a vector with exactly those coordinates.

Derivation: Let \(\mathbf{i}=\langle 1,0,0\rangle\), \(\mathbf{j}=\langle 0,1,0\rangle\), and \(\mathbf{k}=\langle 0,0,1\rangle\). For any three real numbers \(a\), \(b\), and \(c\),

\begin{aligned} a\mathbf{i}+b\mathbf{j}+c\mathbf{k} &=a\langle 1,0,0\rangle+b\langle 0,1,0\rangle+c\langle 0,0,1\rangle\\ &=\langle a,0,0\rangle+\langle 0,b,0\rangle+\langle 0,0,c\rangle\\ &=\langle a,b,c\rangle. \end{aligned}

Thus the coefficients of \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are precisely the first, second, and third coordinates. Conversely, the ordered triple \(\langle a,b,c\rangle\) has those same three coordinates. Applying this fact at each parameter value \(t\), with \(a=x(t)\), \(b=y(t)\), and \(c=z(t)\), proves the stated equivalence of the vector and component forms.

This argument also explains why the order matters. The vector \(\langle a,b,c\rangle\) has first coordinate \(a\), second coordinate \(b\), and third coordinate \(c\). In general, changing the order changes the coordinates and therefore changes the vector.

Keep Coordinate Rules Separate from Cartesian Equations

The equations \(x=x(t)\), \(y=y(t)\), and \(z=z(t)\) are coordinate rules: each coordinate is given as a function of the parameter. They are not, by themselves, a single equation involving only \(x\), \(y\), and \(z\). In particular, the letter \(t\) is part of the parametrized description.

For example, \(x=t\), \(y=t^2\), and \(z=0\) assemble into \(\mathbf{r}(t)=\langle t,t^2,0\rangle\). The expression \(y=t^2\) here gives the \(y\)-coordinate from the parameter; it is not a statement that \(y\) is the parameter. Keeping the parameter and the coordinates distinct helps prevent errors when reading or constructing a vector rule.

Likewise, do not treat an ordered triple as an unordered collection. The rules \(x=t\), \(y=2t\), \(z=3t\) produce \(\langle t,2t,3t\rangle\), not \(\langle 2t,t,3t\rangle\). The latter assigns the first two coordinate formulas to the wrong coordinates.

A Reliable Translation Check

When translating, use the same simple check in either direction: write the three components in \(x\)-, \(y\)-, \(z\)-order and compare them with the coordinate rules. This is a direct check of the defining relationship, not an extra geometric assumption.

Identify the representation.
Determine whether the input is an ordered triple, a unit-vector expansion, a column, or three coordinate equations.

Match coordinates in order.
Pair the first entry with \(x\), the second with \(y\), and the third with \(z\). Keep the same parameter in every component.

Reconstruct and compare.
Write the vector as \(\langle x(t),y(t),z(t)\rangle\) and check that each entry agrees with the given rule.

The check is especially valuable when expressions are visually similar, such as \(2t+1\) and \(t+2\), or when a minus sign appears in only one component. Compare whole expressions, not just their leading terms.

Summary

A parametrized space curve can be written as one vector-valued function or as three coordinate equations. In the vector form \(\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle\), the entries are exactly the coordinate rules. To assemble a vector from coordinate equations, place their right-hand sides in the same \(x\)-, \(y\)-, \(z\)-order. Unit-vector and column notation encode the same ordered components.

Check Your Understanding

Use component order carefully in each question.

Write the coordinate equations for \(\mathbf{r}(t)=\langle 3-t,\ t^2,\ 2t+5\rangle\).
Write a vector-valued function for \(x=4t\), \(y=1-t\), and \(z=t^2+2\).
Convert \(\mathbf{r}(t)=(t^2)\mathbf{i}+(2-t)\mathbf{j}-3\mathbf{k}\) to angle-bracket notation and state its three coordinate rules.
If \(x=t+1\), \(y=2t\), and \(z=5-t\), what is the vector at \(t=3\)? Show each coordinate substitution.
Explain why \(\langle f(t),g(t),h(t)\rangle\) has \(x=f(t)\), \(y=g(t)\), and \(z=h(t)\), rather than a different assignment.