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Unit 3: Lines and Planes
Lesson 2: Vector Equations of Planes
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Plane in 3-space
Requires a
point
on the
plane
and
two non-parallel direction vectors
to specify the
slant
of the plane
Determining the equation of a plane in 3-space
1. Given a point on the plane and
two non-parallel direction vectors
2. Given
three non-collinear points
on the plane
Vector equation of a plane
OP⃗ =
OP0⃗
+
sa⃗
+
tb⃗
where s, t ∈ ℝ
r⃗ =
r0⃗
+ s(
a1
,
a2
,
a3
) + t(
b1
,
b2
,
b3
) where s, t ∈ ℝ
(x, y, z) = (
x0, y0, z0
) + s(
a1,
a2
,
a3
) + t(
b1
,
b2
,
b3
) where s, t ∈ ℝ
Parametric equations
of a plane
x =
x0
+
sa1
+
tb1
y =
y0
+
sa2
+
tb2
z =
z0
+
sa3
+
tb3
where
s
, t ∈ ℝ
