A First Course in Linear Algebra, 2017a

4.6. Parametric Lines 159
This equation determines the line L in R 2 . In fact, it determines a line L in R n . Consider the following
definition.
Definition 4.19: Vector Equation of a Line
Suppose a line L in R n contains the two different points P and P 0 . Let ⃗p and ⃗p 0 be the position
vectors of these two points, respectively. Then, L is the collection of points Q which have the
position vector ⃗q given by
⃗q = ⃗p 0 +t (⃗p − ⃗p 0 )
where t ∈ R.
Let ⃗ d = ⃗p − ⃗p 0 .Then ⃗ d is the direction vector for L and the vector equation for L is given by
⃗p = ⃗p 0 +t ⃗ d,t ∈ R
Note that this definition agrees with the usual notion of a line in two dimensions and so this is consistent
with earlier concepts. Consider now points in R 3 . If a point P ∈ R 3 is given by P =(x,y,z), P 0 ∈ R 3 by
P 0 =(x 0 ,y 0 ,z 0 ), then we can write
⎡
where d ⃗ = ⎣
a
b
c
⎤
⎡
⎣
x
y
z
⎤
⎡
⎦ = ⎣
⎤
x 0
y 0
z 0
⎡
⎦ +t ⎣
⎦. This is the vector equation of L written in component form .
The following theorem claims that such an equation is in fact a line.
Proposition 4.20: Algebraic Description of a Straight Line
Let ⃗a,⃗b ∈ R n with⃗b ≠⃗0. Then⃗x =⃗a +t⃗b, t ∈ R, is a line.
a
b
c
⎤
⎦
Proof. Let ⃗x 1 ,⃗x 2 ∈ R n . Define ⃗x 1 = ⃗a and let ⃗x 2 − ⃗x 1 = ⃗ b. Since ⃗ b ≠⃗0, it follows that ⃗x 2 ≠ ⃗x 1 .Then
⃗a + t⃗b = ⃗x 1 + t (⃗x 2 − ⃗x 1 ). It follows that ⃗x = ⃗a + t⃗b is a line containing the two different points X 1 and X 2
whose position vectors are given by⃗x 1 and⃗x 2 respectively.
♠
We can use the above discussion to find the equation of a line when given two distinct points. Consider
the following example.
Example 4.21: A Line From Two Points
Find a vector equation for the line through the points P 0 =(1,2,0) and P =(2,−4,6).
Solution. We will use the definition of a line given above in Definition 4.19 to write this line in the form
⃗q = ⃗p 0 +t (⃗p − ⃗p 0 )