Linear Algebra, 2020a

Section II. Linear Geometry 39
is the one shown in the picture as having its whole body in the line — it is a
direction vector for the line. Note that points on the line to the left of x = 1
are described using negative values of t.
In R 3 , the line through (1, 2, 1) and (0, 3, 2) is the set of (endpoints of) vectors
of this form
z
⎛ ⎞ ⎛ ⎞
1 −1
{ ⎝2⎠ + t · ⎝ 1 ⎠ | t ∈ R}
1 1
y
x
and lines in even higher-dimensional spaces work in the same way.
In R 3 , a line uses one parameter so that a particle on that line would be
free to move back and forth in one dimension. A plane involves two parameters.
For example, the plane through the points (1, 0, 5), (2, 1, −3), and (−2, 4, 0.5)
consists of (endpoints of) the vectors in this set.
⎛
⎜
1
⎞ ⎛ ⎞ ⎛ ⎞
1 −3
⎟ ⎜ ⎟ ⎜ ⎟
{ ⎝0⎠ + t ⎝ 1 ⎠ + s ⎝ 4 ⎠ | t, s ∈ R}
5 −8 −4.5
The column vectors associated with the parameters come from these calculations.
⎛ ⎞ ⎛ ⎞ ⎛
1 2
⎜ ⎟ ⎜ ⎟ ⎜
1
⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
−3 −2 1
⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ 1 ⎠ = ⎝ 1 ⎠ − ⎝0⎠
⎝ 4 ⎠ = ⎝ 4 ⎠ − ⎝0⎠
−8 −3 5 −4.5 0.5 5
As with the line, note that we describe some points in this plane with negative
t’s or negative s’s or both.
Calculus books often describe a plane by using a single linear equation.
⎛ ⎞
x
P = { ⎝y⎠ | 2x + y + z = 4}
z
To translate from this to the vector description, think of this as a one-equation