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The position vector is a fundamental concept in understanding curves in three dimensions. Imagine it as a mathematical arrow pointing from the origin to a specific point on the curve. This vector is denoted as \( \mathbf{r}(t) \). It depends on a parameter \( t \), which typically represents time or a variable along the curve.

For example, if we take \( \mathbf{r}(t) = \langle \cos t, \sin t, \cos(6t) \rangle \), each component \( \cos t, \sin t, \) and \( \cos(6t) \) describes how each coordinate changes as \( t \) changes. This means at each value of \( t \), you can plot a point on the curve in 3D space, forming a path traced by these points.

Understanding the position vector helps us set the stage for further operations like finding the tangent line, which gives us a direction of motion at any point on the curve.

Parametric equations are a useful way to express a curve using a parameter, typically \( t \). Instead of describing \( y \) as a function of \( x \) alone, parametric equations use a third variable to express both \( x \) and \( y \) (and \( z \), in 3D). This gives more flexibility, especially when modeling curves that are difficult to represent with just \( y = f(x) \).

In our example, the tangent line is expressed using parametric equations. They take the form:

• \( x(t) = \frac{\sqrt{2}}{2} - t\frac{\sqrt{2}}{2} \)
• \( y(t) = \frac{\sqrt{2}}{2} + t\frac{\sqrt{2}}{2} \)
• \( z(t) = 6t \)

Here, each equation tells us the coordinates of a point on the tangent line as \( t \) varies. When \( t = 0 \), the equations give you the point where the tangent touches the curve. As \( t \) changes, the equations describe points further along the tangent line.

Vector differentiation is the process of finding the derivative of a vector function, like our position vector \( \mathbf{r}(t) \). This derivative, \( \mathbf{r}'(t) \), is called the velocity vector, and it points in the direction of the curve's tangent at a given point.

To differentiate the position vector \( \langle \cos t, \sin t, \cos(6t) \rangle \), we take the derivative of each component separately:

• The derivative of \( \cos t \) is \( -\sin t \).
• The derivative of \( \sin t \) is \( \cos t \).
• The derivative of \( \cos(6t) \) is \( -6\sin(6t) \), applying the chain rule.

The vector derivative \( \mathbf{r}'(t) = \langle -\sin t, \cos t, -6\sin(6t) \rangle \) provides the direction of the tangent line. It's key to parametric equations of the tangent line, as it informs the direction in which \( t \) increases or decreases, building the line through a specific point on the curve.