Math 214 Lecture 9
Lines in R3
Desmos app to visualize lines.
Math 214 Lecture 9
Parametric Equations
The vector form of the equation of a line ℓ through point P = (x0 , y0 , z0 ) and in the direction d = ⟨dx , dy , dz ⟩ is
given by:
r = p + td
The parametric equations of ℓ are the corresponding component functions:
x = x0 + tdx
y = y0 + tdy
z = z0 + tdz
Example. Find an equation of the line that goes the points (1,1,2) and (3,2,4)
Math 214 Lecture 9
Planes in R3
Desmos app to visualize planes and normal vectors.
Math 214 Lecture 9
Definition. The Equation of a Plane through point P = (x0 , y0 , z0 ) with normal vector n = ⟨a, b, c⟩
n · (x − P ) = 0
⟨a, b, c⟩ · ⟨x − x0 , y − y0 , z − z0 ⟩ = 0
a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0
ax + by + cz = d
where d = ax0 + by0 + cz0 .
Example. Find an equation of the plane through (1,2,-3) with a normal vector ⟨1, −2, 4⟩. Sketch the plane.
Math 214 Lecture 9
Example. Find an equation of the line that is the intersection of the planes x − y = 3 and x + y + z = 2.
Math 214 Lecture 9
What is a function?
Math 214 Lecture 9
Definition. Let J be an interval of R. A function f : J → Rn is called a vector-valued function.
When n = 2, its image is normally referred to as a plane curve (because it lies on a plane).
When n = 3, its image is referred to as a space curve (because it’s in “space”).
When J is a closed interval, the point γ(a) is called the initial point, and the point γ(b) is called the end or terminal
point.
Math 214 Lecture 9
Example. Sketch and identify the curve defined by the parametric equations
x = t2 − 2t,
y =t+1
Math 214 Lecture 9
Finding parametric equations to describe a curve is not always easy, but sometimes, we define a curve by simply using parametric
equations to begin with.
Example. Describe the motion of the particle whose coordinates at time t are
x = cos t and y = sin t
Math 214 Lecture 9
Example. Describe the motion of the particle whose coordinates at time t are given by
⃗r(t) = ⟨cos t, sin t, t⟩
Math 214 Lecture 9
Example. Describe the motion of the particle whose coordinates at time t are given by
⃗r(t) = ⟨cos t, sin t, sin t⟩
Math 214 Lecture 9
The notions of limits, continuity, and differentiation all have very natural analogues for vector-valued functions.
Definition. For a vector-valued function r defined on an open interval containing t = a, we will often write
⃗ ∈ Rn
lim r(t) = L
t→a
⃗ whenever the value of t is close enough to a, without
if the value of the function r(t) is sure to be arbitrarily close to L
being exactly t.
Definition. A vector-valued function r(t) is continuous at a if
lim r(t) = r(a)
t→a
Math 214 Lecture 9
Definition. The derivative of the vector-valued function r(t) is defined to be
r′ (t) =
r(t + h) − r(t)
dr
(t) = lim
h→0
dt
h
when the limit exists.
The vector r′ (t0 ) is called the tangent vector to the curve C defined by r at the point r(t0 ), provided that it is not 0.
The tangent line to the curve C at the point r(t0 ) is defined to be the line through the point r(t0 ) and parallel to r′ (t0 ).
It is often convenient to work with′ a unit vector in the same direction as the tangent vector, this is called the unit tangent
vector and is defined by T(t) = ∥rr′ (t)
.
(t)∥
Math 214 Lecture 9
Example. Compute the derivatiave of r(t) = cos(2t) î + tet ĵ + lnt t k̂