Straight lines in 3d

Problem

The straight lines \(g: \vec{X} = \begin{pmatrix}8\\1\\7\end{pmatrix} + \lambda\cdot \begin{pmatrix}3\\1\\2\end{pmatrix}\), \(\lambda \in \mathbb{R}\), and \(h: \vec{X} = \begin{pmatrix}-1\\5\\-9\end{pmatrix} + \mu \cdot \begin{pmatrix}1\\-2\\4\end{pmatrix}\), \(\mu \in \mathbb{R}\), are given in a cartesian coordinate system. The lines \(g\) and \(h\) intersect at the point \(T\).

1. Determine the coordinates of \(T\).

   (result: \(T(2|-1|3)\))

2. Give the coordinates of two points \(P\) and \(Q\) which lie on \(g\) and are at equal distance from \(T\).

3. Two points \(U\) and \(V\) of the line \(h\) together with the points \(P\) and \(Q\) form the rectangle \(PUQV\). Describe a method to determine the coordinates of \(U\) and \(V\).

Solution of part a

One obtains the intersection of the lines \(g\) and \(h\) by equating the equations of the two lines. The equations for the three coordinates yield the system of linear equations

\[\begin{split}3\lambda -\mu &= -9\\ \lambda+2\mu &= 4\\ 2\lambda-4\mu &= -16\end{split}\]

with the solution \(\lambda=-2\), \(\mu=3\). Plugging this into the equation of the lines yields the given coordinates of the point \(T\).

The solution can be determined with the help of Sage:

To illustrate the result one can draw the lines in a three-dimensional coordinate system with Sage.

Solution of part b

The intersection \(T\) can be obtained by plugging the value of \(\lambda\) determined in part a into the equation of line \(g\). Now, two points \(P\) and \(Q\) are to be determined which lie on \(g\) at equal distance from \(T\). To this end, one adds a chosen value to \(\lambda\) and subtracts the same value from \(\lambda\).

In part a, we found \(\lambda = -2\). The two points \(P\) and \(Q\) can be obtained, for example, by plugging the values \(\lambda = -1\) and \(\lambda = -3\) into the equations of the line. These points are added to the coordinate system.

Solution of part c

In the following, a method is described for finding two points \(U\) and \(V\) which lie on line \(h\) such that \(PUQV\) forms a rectangle.

The points \(P\) and \(Q\) are opposite corners of the rectangle. It follows that the line \(g\) between these points is a diagonal of the rectangle. Since \(T\) is exactly the center between those two points, it has to be the center of the rectangle as well.

The two other points shall lie on the line \(h\). The same considerations as before imply that the line \(h\) between the points \(U\) and \(V\) is the second diagonal of the rectangle.

To obtain a rectangle from a quadrangle with known diagonals, the diagonals have to have the same length and the center of the diagonals has to be the intersection of these. Thus, the two points \(U\) and \(V\) have to be at equal distance from \(T\), just as the points \(P\) and \(Q\).

With Sage, we first compute the distance between the points \(T\) and \(Q\). Subsequently, the value \(\mu\) is determined for which the corresponding point on the line \(h\) is at equal distance from \(T\) as the point \(Q\). Then, the points \(U\) and \(V\) can be obtained by plugging \(\pm\mu\) into the equation of line \(h\).