What is meant by parameter in this context?

Question

A part of my homework question is "Let $L_0$ be the line $(x, y, z) = (2, −1, 6) + u(1, 2, 0)$ (where $u$ is a parameter)." What does it mean by parameter in this context?

Answer 1

The word 'parameter' has (at least) $2$ meanings in mathematics:

1. Parameters can be constants which specify a class of different functions. For instance, the general equation of an ellipse is $$\left(\frac xa\right)^2 + \left(\frac yb\right)^2 = 1$$ Here the $a$ and $b$ are the parameters: they are considered constants, but they can take on the value of any positive real number.
2. Parameterizing a curve (/surface/ manifold/ $\dots$) is the process of introducing one or more variables that don't correspond to the axes of our space to make it easier to write explicitly. For example, look again at the general equation of an ellipse. You can see that it is written in implicit form. What if we introduced the parameter $t$ and set it equal to $x$? Then we could specify the curve parametrically as $(x,y) = (t, \pm b\sqrt{1-(t/a)^2})$. This may not seem necessary to you, because we could have just written the function explicitly as $y=\pm b\sqrt{1-(x/a)^2}$, but in more than $2$ dimensions, it is not always possible to write an equation for a curve as $y=f(x,z)$ or $z=g(x,y)$. For instance, try to solve your HW problem for $z$ in terms of $x$ and $y$.

The second meaning is the one used in this problem.

Answer 2

We set: $$X(u) = \left( {\begin{array}{*{20}{c}} {x(u)}\\ {y(u)}\\ {z(u)} \end{array}} \right),P = \left( {\begin{array}{*{20}{c}} 2\\ { - 1}\\ 6 \end{array}} \right),A = \left( {\begin{array}{*{20}{c}} 1\\ 2\\ 0 \end{array}} \right)$$

then our line L is given by: $$L:X(u) = P + u \cdot A$$

here, $P$ is our starting-point, $A$ is our direction-vector, and our parameter $u$ is used to move on the line. Our parameter $u$ may have values $ - \infty < u < \infty $

For negative values we change our direction, we move backward. For $u=0$ we are at our starting point $P$.

The line is given in a parametric form. A line in ${R^3}$ could also be defined as section of two planes in ${R^3}$, but then we need two planes that inersect, that means also two equations for planes. In our case here we need only one equation.

From our parametrization $$\left( {\begin{array}{*{20}{c}} x\\ y\\ z \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {u + 2}\\ {2u - 1}\\ 6 \end{array}} \right)$$

we get the system

$$\begin{array}{l} x = u + 2\\ y = 2u - 1\\ z = 6 \end{array}$$

Solving for $u$ we get:

$$u = x - 2$$

and from the other two equations, we have:

$$\begin{array}{l} I:2x - y - 5 = 0\\ II:z - 6 = 0 \end{array}$$

These are just two equations for two planes in ${R^3}$, which intersect along the line.