A First Course in Linear Algebra, 2017a

162 R n
Let t = x−2 y−1
3
,t =
2
and t = z + 3, as given in the symmetric form of the line. Then solving for x,y,z,
yields
⎫
x = 2 + 3t ⎬
y = 1 + 2t
⎭
with t ∈ R
z = −3 +t
This is the parametric equation for this line.
Now, we want to write this line in the form given by Definition 4.19. This is the form
⃗p = ⃗p 0 +t ⃗ d
where t ∈ R. This equation becomes
⎡
⎣
x
y
z
⎤
⎡
⎦ = ⎣
2
1
−3
⎤
⎡
⎦ +t ⎣
3
2
1
⎤
⎦, t ∈ R
♠
Exercises
Exercise 4.6.1 Find the vector equation for the line through (−7,6,0) and (−1,1,4). Then, find the
parametric equations for this line.
Exercise ⎡ ⎤4.6.2 Find parametric equations for the line through the point (7,7,1) with a direction vector
1
⃗d = ⎣ 6 ⎦.
2
Exercise 4.6.3 Parametric equations of the line are
x = t + 2
y = 6 − 3t
z = −t − 6
Find a direction vector for the line and a point on the line.
Exercise 4.6.4 Find the vector equation for the line through the two points (−5,5,1), (2,2,4). Then, find
the parametric equations.
Exercise 4.6.5 The equation of a line in two dimensions is written as y = x−5. Find parametric equations
forthisline.
Exercise 4.6.6 Find parametric equations for the line through (6,5,−2) and (5,1,2).
Exercise 4.6.7 Find the vector ⎡ equation ⎤ and parametric equations for the line through the point (−7,10,−6)
with a direction vector d ⃗ 1
= ⎣ 1 ⎦.
3