91Ó°ÊÓ

When finding the parametric equations of a line, it's essential to understand the concept of a direction vector. A direction vector indicates the line's direction in space and is represented by components along the x, y, and z axes. In the context of a three-dimensional line, the direction vector is \( \vec{d} = (a, b, c) \). Each component, (a, b, c), tells us how much the line moves in each respective axis with a change in a parameter (usually represented by \( t \)).

For a line parallel to a particular axis, the direction vector aligns with that axis, meaning only the component related to that axis will be non-zero. For example, in our problem, since the line is parallel to the z-axis, the direction vector \( (0, 0, 1) \) means the line doesn't shift in either the x or y directions but moves up or down along the z-axis. This simplifying knowledge helps in constructing reliable parametric equations.

Parametric equations describe the position of any point on the line based on a parameter \( t \). The general form utilizes the direction vector and a known point on the line (denoted as \( (x_0, y_0, z_0) \)). The parametric form of a line is given as:

• \( x = x_0 + at \)
• \( y = y_0 + bt \)
• \( z = z_0 + ct \)

In the given exercise, we know the line passes through the point \( (1, 1, 1) \) and has a direction vector \( (0, 0, 1) \). Thus, our equations become:

• \( x = 1 + 0t = 1 \)
• \( y = 1 + 0t = 1 \)
• \( z = 1 + 1t = 1 + t \)

These equations allow us to find any point along the line by substituting different values of \( t \). Here, both x and y are constant, simplifying visualization since the line only moves in the z-direction.

When a line is described as being parallel to the z-axis, it means that the line runs exactly in the direction of z and doesn't shift in the x or y directions. In simpler terms, it's as if the line is "standing up" along the steel column of the z-axis, untouched by any leanings towards the x or y directions.

In mathematical terms, this parallelism is represented by a direction vector like the one seen here: \( (0, 0, 1) \). This vector clearly indicates no movement along x and y, reinforcing that the path adheres strictly to changes in the z-direction. A practical visualization might be imagining a line that passes through a specific point which continues vertically aligned along z, marking a straightforward, upright path that rises or falls like an elevator.