91Ó°ÊÓ

In the given problem, we start with the concept of a vector equation which is crucial for understanding curves in three-dimensional space. A vector equation is represented as \( \mathbf{r}(t) = t \mathbf{i} + t \mathbf{j} + t^2 \mathbf{k} \). Here, \( t \) is a parameter that changes over time, and each component (\( \mathbf{i}, \mathbf{j}, \mathbf{k} \)) represents the axis directions in 3D space respectively for the x, y, and z coordinates.

- The component along \( \mathbf{i} \) is \( x(t) = t \)- The component along \( \mathbf{j} \) is \( y(t) = t \)- The component along \( \mathbf{k} \) is \( z(t) = t^2 \)

These equations show how a point moves in space, determining a detailed pathway known as a **curve**. Here, each component reflects a critical aspect of the parabola's path in 3D.

Parametric equations involve expressing the coordinates of points on a curve as functions of a single variable, commonly denoted as \( t \). They play a dominant role in describing complex curves in space, like parabolas. In this exercise, the parametric equations are:\[\begin{align*}x(t) &= t \y(t) &= t \z(t) &= t^2\end{align*}\]

These parametric equations allow us to reveal the relationship among \( x \), \( y \), and \( z \) by substituting \( t \) into each equation. Notice both \( x \) and \( y \) equal \( t \), leading to \( z = t^2 \).

Substituting these gives two similar equations:\[ z = x^2 \z = y^2 \]

This equivalence shows that whether you substitute \( t \) in terms of \( x \) or \( y \), it unveils the parabolic characteristic \( z = x^2 \) or \( z = y^2 \), indicating that the path is indeed a parabola which extends along the z-axis.

In mathematics, the focus of a parabola is one of its defining features. It's a special point about which the parabola curves. Particularly in 3D, the parabola represented by \( z = x^2 \) is a parabolic cylinder opening along the z-axis. To locate the focus of this 3D parabola, we refer to its standard form seen in an equation like \( z = 4px^2 \).

Here, the focus is found using the formula \( z = \frac{1}{4p} \), specific to standard parabolas in 2D. For the given equation ({\( z = x^2 \)}, a form equatable to \( z = 4px^2 \)) the value of \( p \) comes out as 1 when considering parabolas in typical vertex form. Thus,\[Focus: \left(0, 0, \frac{1}{4}\right)\]

This calculation helps in understanding the key position of the focus as \( (0, 0, \frac{1}{4}) \), crucial for defining the parabola's geometric properties and determining the pathway of the curve. This focus outlines where the parabola curves towards in its three-dimensional space.