Lecture Note - Planes in Three-Space
Michael Wang
1. Consider a plance P that passes through a point P0 = (x0 , y0 , z0 ). We can determine P completely
by specifying a nonzero vector n = ha, b, ci that is orthogonal to P. Such a vector is called a normal
vector.
Theorem 1. (Equation of a Plane). Plane through P0 = (x0 , y0 , z0 ) with normal vector n = ha, b, ci
n · hx, y, zi = d
(1)
a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0
(2)
ax + by + cz = d
(3)
Vector form:
Scalar forms:
where d = n · hx0 , y0 , z0 i = ax0 + by0 + cz0 .
Note that if n is normal to plane P, then so is every nonzero scalar multiple λn. When we use λn
instead of n, the resulting equation for P changes by a factor of λ.
2. Parallel Planes
Two planes P and P 0 are parallel if they have a common normal vector. In general, a family of
parallel places is obtained by choosing a normal vector n = ha, b, ci and varying the constant d in the
equation.
ax + by + cz = d
The unique place in this family through the origin has equation ax + by + cz = 0.
Points that lie on a line are called collinear. If we are given three points P, Q, and R that are not
collinear, then there is just one plane passing through P, Q, and R.
3. The Plane Determined by Three Points
Let us say, we have three points P = ha1 , b1 , c1 i, Q = ha2 , b2 , c2 i, and R = ha3 , b3 , c3 i. To find the
plane determined by these three points, we need to find the normal vector, which is determined by
−−→
−→
the cross product of P Q and P R.
i
j
k −−→ −→ n = P Q × P R = a2 − a1 b2 − b1 c2 − c1 a3 − a1 b3 − b1 c3 − c1 The we need to choose a point on the plane, say P , and compute d.
−−→
d = n · OP
Then we can find out the plane.
4. Intersection of a Plane and a Line
Find the point P where the plane ax + by + cz = d and the line r(t) = hx0 , y0 , z0 i + tha1 , b1 , c1 i
intersect.
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The one has parametric equations:
x = x0 + a1 t
y = y0 + b1 t
z = z0 + c1 t
Substitute in the equation of the plane and solve for t:
ax + by + cz = a(x0 + a1 t) + b(y0 + b1 t) + c(z0 + c1 t) = d
Therefore, we can find the P coordinates.
The intersection of a plane P with a coordinate plane or a plane parallel to coordinate plane is called
a trace. The trace is a line unless P is parallel to the coordinate plane.
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