Question

Show that every line parallel to the \(z\) axis has parametric equations \(x=x_{0}, y=y_{0}, z=t\) for some fixed numbers \(x_{0}\) and \(y_{0}\).

Short answer

Lines parallel to the \(z\)-axis have parametric equations \(x = x_0, y = y_0, z = t\).

Step-by-step solution

Understanding the Problem

We need to show that any line parallel to the \(z\)-axis can be represented with the given parametric equations: \(x = x_0, y = y_0, z = t\). Here, \(x_0\) and \(y_0\) are constants, and \(t\) is a parameter that changes.

Identifying Line Properties

A line parallel to the \(z\)-axis means that it does not change its position along the \(x\) and \(y\) axes. This implies that the direction vector for a line parallel to the \(z\)-axis is \((0, 0, 1)\), with only the \(z\) component changing.

Express the Line Equations

For any line in space, a parametric equation is given by starting point plus a parameter times the direction vector. In this case:- A point on the line can be \((x_0, y_0, z_0)\), where \(x_0\) and \(y_0\) are constants and \(z_0\) is some initial \(z\)-value.- The direction vector is \((0, 0, 1)\).

Derive Parametric Equations

Using the point \((x_0, y_0, z_0)\) and the direction \((0, 0, 1)\), the line equations become: \[x = x_0 + 0 \cdot t = x_0\]\[y = y_0 + 0 \cdot t = y_0\]\[z = z_0 + 1 \cdot t\]We often shift parameters so \(z = t\) to simplify, making initial \(z_0\)-value irrelevant.

Conclusion

The derived parametric equations are \(x = x_0\), \(y = y_0\), and \(z = t\), confirming the expression \(x=x_{0}, y=y_{0}, z=t\), for any fixed \(x_0\) and \(y_0\), indeed represents every line parallel to the \(z\)-axis.

Key concepts

Direction Vector

When you're looking at a line in a 3D space, the direction vector tells you which direction the line is going. Essentially, it's like an arrow showing the path of the line. For example, if a line is parallel to the \(z\)-axis, its direction vector is \((0, 0, 1)\). This means that the line moves upwards or downwards along the \(z\)-axis, but stays in the same spot on the \(x\) and \(y\) axes.

Direction vectors are crucial for writing parametric equations of lines. They help show which part of the point on the line changes with the parameter \(t\). In our case, the parameter \(t\) affects only the \(z\)-coordinate, not \(x\) or \(y\), because the direction vector is precisely \((0, 0, 1)\), indicating no change in those dimensions.

Lines in Space

Lines in space are like invisible threads that connect points in a three-dimensional universe. They're mathematically represented using parametric equations. These equations express each of the coordinates \(x\), \(y\), and \(z\) in terms of a parameter \(t\).

Imagine starting at a point \((x_0, y_0, z_0)\). The parametric equations are set up as:

• \(x = x_0 + a\cdot t\)
• \(y = y_0 + b\cdot t\)
• \(z = z_0 + c\cdot t\)

where \(a, b, c\) are the components of the direction vector. In simple terms, each line is drawn out by adjusting \(t\), like pulling on a string to move from one bead to the next.

For lines parallel to the \(z\)-axis, the direction vector \((0, 0, 1)\) simplifies these equations to \(x = x_0\), \(y = y_0\), and \(z = t\). This shows how parametric equations dynamically describe the line's behavior in space.

Parallel Lines

Parallel lines are like twin universes that never touch, no matter how far you extend them in any direction. In both two-dimensional and three-dimensional spaces, these lines remain equidistant from each other.

To identify parallel lines in 3D, you check their direction vectors. If two lines have the same direction vector, they are parallel. For instance, any two lines parallel to the \(z\)-axis both have the direction vector \((0, 0, 1)\). This commonality ensures that while they may lie at different starting points – different \(x_0\) and \(y_0\) – they stretch out in the same direction, confirming their parallel nature.

By understanding the role of direction vectors and parametric equations, we can easily identify, represent, and explore not just parallel lines but many other spatial relationships slightly more complex than typical line drawings you might be used to from 2D geometry.