Discover Linear Algebra

Subsection 15.3.1 Lines in the plane

When we view a single linear equation in two variables as a (very simple) system of equations, we require a single parameter to solve. We've previously seen that we can use matrix algebra to express the general solution to a system of equations as a linear combination of column matrices, where the parameters appear as coefficients.

When we interpret the column matrices in such a linear combination as vectors, we can investigate the geometry of the set of solutions, as we did in Discovery 15.1. For a general solution to \(ax+by=c\) of the form

the vector \(\uvec{x}_0\) corresponds to the particular solution for \(t=0\text{,}\) and we can think of its terminal point as an “base” point on the line. When we vary the value of the parameter \(t\text{,}\) we get solutions that are vector sums of the base point \(\uvec{x}_0\) and scalar multiples of \(\uvec{p}\text{.}\) Geometrically, these vector sums all involve tacking some scaled copy of \(\uvec{p}\) onto the end of \(\uvec{x}_0\text{.}\)

The terminal points of all such vectors \(\uvec{x}\) trace out the line parallel to \(\uvec{p}\) that passes through the terminal point of \(\uvec{x}_0\text{.}\) Since \(\uvec{p}\) is parallel to the line, we might think of \(\uvec{p}\) as a “direction” vector for the line.

Subsection 15.3.2 Lines in space

Two nonparallel planes in space must intersect in a line, as in Discovery 15.3. If we have algebraic equations \(a_1 x + b_1 y + c_1 z = d_1\) and \(a_2 x + b_2 y + c_2 z = d_2\) for these planes, then solving for the points of intersection is the same as solving the linear system formed by these two equations. The assumption that the planes are not parallel guarantees that we will need one (and only one) parameter to solve the system, and then the general solution can be expressed in a vector form

just as in the previous case of a line in the plane. To visualize, we can imagine the diagram in the previous subsection above as floating in space instead of lying in the plane.

Subsection 15.3.3 Planes in space

When we view a single linear equation in three variables as a system of equations, we require two parameters to solve. As before, we can use matrix algebra to express the general solution as a linear combination of column matrices, where the parameters appear as coefficients.

Similarly to the vector description of a line, a parametric vector expression

can be interpreted as follows. The terminal point of the vector \(\uvec{x}_0\) is an “base” point on the plane, corresponding to parameter values \(s=0\) and \(t=0\text{.}\) The vectors \(\uvec{p}_1\) and \(\uvec{p}_2\) are parallel to the plane. Similarly to the vector description of a line, as we vary the values of \(s\) and \(t\) we obtain other points on the plane by tacking on linear combinations of \(\uvec{p}_1\) and \(\uvec{p}_2\) to the end of \(\uvec{x}_0\text{.}\)

Subsection 15.3.4 Parallel vectors as a “basis” for lines and planes

In a vector description \(\uvec{x} = \uvec{x}_0 + t \uvec{p}\) for a line, the “base” point at the head of \(\uvec{x}_0\) gets you onto the line, and then one can get to any other point on the line by following a scalar multiple of the parallel vector \(\uvec{p}\text{.}\) In this way, the parameter \(t\) effectively places a coordinate system on the line, where the integers are spaced apart by the length of \(\uvec{p}\text{.}\)

Values of the parameter \(t\) are mapped to specific positions on the line, just as when we visualize the set of real numbers \(\R\) along the real number line, where each real number represents a position on a line. This idea of a coordinate system along the line is more natural when the line passes through the origin, so that we can take \(\uvec{x}_0 = \zerovec\text{.}\) In this case we have \(\uvec{x} = t \uvec{p}\text{,}\) so that all points on the line correspond to scalar multiples of the parallel vector \(\uvec{p}\text{,}\) and parameter value \(t=0\) corresponds to the origin. So the vector \(\uvec{p}\) tells us pretty much all we need to know about the line, and any other line that is parallel to this line could use the same parallel vector \(\uvec{p}\text{,}\) it would just need a different “base point” vector \(\uvec{x}_0\text{.}\)

In the plane, the standard basis vectors \(\uvec{e}_1\text{,}\)\(\uvec{e}_2\) play the same role for the whole plane, representing our \(xy\)-coordinate system and setting up a grid as in Discovery 15.5.

When we have a vector description \(\uvec{x} = \uvec{x}_0 + s \uvec{p}_1 + t \uvec{p}_2\) for a plane in space, scalar multiples of the vectors \(\uvec{p}_1\) and \(\uvec{p}_2\) form a grid on the plane in the same way (except that the grid lines will not be at right angles to each other if \(\uvec{p}_1\) and \(\uvec{p}_2\) are not).

The vectors \(\uvec{p}_1\) and \(\uvec{p}_2\) set up an \(st\)-coordinate system on the plane, where every point on the plane corresponds to a particular pair of parameter values, and vice versa, by adding the linear combination \(s \uvec{p}_1 + t \uvec{p}_2\) onto \(\uvec{x}_0\text{.}\) If the plane passes through the origin (as in Discovery 15.7), then we can take \(\uvec{x}_0\) to be the zero vector, so that the origin corresponds to \((s,t) = (0,0)\text{.}\) Then every other point in the plane could be constructed as a linear combination of \(\uvec{p}_1\) and \(\uvec{p}_2\text{.}\)

Subsection 15.3.5 Summary

Combining with our knowledge of normal vectors from the previous chapter, we now have several ways to describe lines and planes in \(\R^2\) and \(\R^3\text{.}\)