Michael Corral: Vector Calculus

1.5 Lines and Planes 31
1.5 Lines and Planes
Now that we know how to perform some operations on vectors, we can start to deal
withsomefamiliargeometricobjects, likelinesandplanes, inthelanguageofvectors.
The reason for doing this is simple: using vectors makes it easier to study objects in
3-dimensional Euclidean space. We will first consider lines.
Line through a point, parallel to a vector
Let P=(x 0 ,y 0 ,z 0 ) be a point in 3 , let v=(a,b,c) be a nonzero vector, and let L be the
line through P which is parallel to v (see Figure 1.5.1).
r+tv
t0
y
L
x
Figure 1.5.1
Let r=(x 0 ,y 0 ,z 0 ) be the vector pointing from the origin to P. Since multiplying the
vector v by a scalar t lengthens or shrinks v while preserving its direction if t>0, and
reversing its direction if t