In this lesson, we will discuss lines in space. Our goal is to generalize what we've learned about lines in the plane. The first approach we used in the plane involved the concept of slope. However, in three-dimensional space, we cannot characterize slope with a single number, because changing the x-coordinate affects not only the y-coordinate but also the z-coordinate, and not necessarily to the same extent. Moreover, the number of special cases to consider also increases. Therefore, we will not generalize this particular approach.

In the second approach, we defined the line ee using one of its points and a direction vector. This approach can easily be generalized to three-dimensional space. The only difference is that now we have three coordinates instead of two. Thus, the line e is given by a point P₀(x₀;y₀;z₀) lying on it, and a direction vector v(v₁;v₂;v₃) ≠ 0.

However, the reasoning remains the same: a point P(x;y;z) plies on the line e, if and only if the vector P₀P is parallel to v.

In coordinate form:

This is called a parametric system of equations of the line e. Once again, it's true that the parametric equations depend on the choice of the point P₀ and the direction vector v.

Example1 

Now let's determine a parametric system of equations for the line e passing through the point P₀(5;-4;2) with direction vector v(3;2;-3). All we need to do is substitute:

Example2 

Now let's consider the system of equations for the line ee passing through the point P₀(-4;5;-1) with direction vector v(0;0;1). Substituting the given data, we obtain the following system of equations. Since now v = k, meaning the line is parallel to the z-axis, both x and y remain constant. The third equation merely indicates that z can take all real values.

In the general case, we can again express the parameter from each of the equations.

Of course, this transformation can only be performed if none of the coordinates of v is zero (that is, if the line is not parallel to any of the coordinate planes). Since the value of the parameter is actually irrelevant to us, from this form we obtain a parameter-free system of equations for the line.

If two coordinates are zero, for example v₁ = v₂ = 0, then the line is parallel to one of the coordinate axes – in this case, the z-axis. In such cases, two coordinates of the points on the line remain constant (here, the x- and y-coordinates), while the third coordinate takes all real values. Thus, the system of equations of the line can take one of the following three forms (in our example, the first variant applies).