91Ó°ÊÓ

When dealing with parametric curves, determining if they lie on a plane is a common problem. For the curve \( \mathbf{r}(t) = 2t \mathbf{i} + t^2 \mathbf{j} + (1 - t^2) \mathbf{k} \), we can express each component \( x(t) \), \( y(t) \), and \( z(t) \) to reveal the relation between them. Here:

• \( x(t) = 2t \)
• \( y(t) = t^2 \)
• \( z(t) = 1 - t^2 \)

Notice that both \( y(t) \) and \( z(t) \) depend on \( t^2 \), allowing us to relate them directly: \( z + y = 1 \). This equation \( z + y = 1 \) defines a plane in three-dimensional space showing that at any value of \( t \), the curve lies on this plane.

The tangent line of a parametric curve at a specific point provides insights into its behavior at that point. To find the tangent line to our curve \( \mathbf{r}(t) = 2t \mathbf{i} + t^2 \mathbf{j} + (1 - t^2) \mathbf{k} \) at \( t = 2 \), we begin by calculating its derivative, \( \mathbf{r}'(t) \). This derivative gives the direction of the tangent to the curve:

• \( \mathbf{r}'(t) = 2\mathbf{i} + 2t\mathbf{j} - 2t\mathbf{k} \)

Plugging \( t = 2 \) into \( \mathbf{r}'(t) \) yields \( \mathbf{r}'(2) = 2\mathbf{i} + 4\mathbf{j} - 4\mathbf{k} \). The position at this point is \( \mathbf{r}(2) = 4\mathbf{i} + 4\mathbf{j} - 3\mathbf{k} \). Therefore, the tangent line \( \mathbf{L}(s) \) combining this direction and position can be expressed as:\[ \mathbf{L}(s) = (4 + 2s)\mathbf{i} + (4 + 4s)\mathbf{j} + (-3 - 4s)\mathbf{k} \].

Finding where the tangent line intersects with the \( xy \)-plane involves setting its \( z \)-coordinate to 0. This is because the \( xy \)-plane is described by \( z = 0 \). For our tangent line \( \mathbf{L}(s) = (4 + 2s)\mathbf{i} + (4 + 4s)\mathbf{j} + (-3 - 4s)\mathbf{k} \), we solve:

• \(-3 - 4s = 0 \)

Solving this equation, we find \( s = -\frac{3}{4} \). Substituting this value back into \( x \) and \( y \) gives:

• \( x = 4 + 2(-\frac{3}{4}) = 2.5 \)
• \( y = 4 + 4(-\frac{3}{4}) = 1 \)

Thus, the point of intersection with the \( xy \)-plane is \( (2.5, 1, 0) \), which provides valuable information about where this tangent line reaches the plane.