12.7 Planes
Let N = Ai + Bj + Ck be the nonzero vector perpendicular to a plane through point P and let Q be
another point on the plane as shown:
So, PQ =
Now, since N is perpendicular to the plane, N i PQ = 0. Using this we can say the equation of the
plane is:
Examples:
1) Find an equation for the plane which passes through the point P(1, 2, -3) and is perpendicular to
i – j + 2k
2) Find an equation for the plane which passes through the point P(1, 2, -3) and is parallel to the
plane x – 2y + z -3 = 0
Unit normals to a plane:
uN =
**If N is a normal to a plane then all scalar multiples of N are normals to that plane.
Intersections:
Given two planes P1 and P2, then exactly one of the following holds:
1. P1 and P2 are parallel
2. P1 and P2 coincide
3. P1 and P2 intersect in a straight line
Examples:
Put 4x + 5y – 6z = 60 in intercept form and find the intercepts.
Determine if the following vectors are coplanar: j – k, 3i – j + 2k, 3i – 2j + 3k
Angle between two planes: cos θ = u N 2 • u N 2 =
Example:
Find the angle between the planes:
4x + 4y – 2z = 3
and
2x + y + z = -1
Finding planes given 3 points:
Find%an%equation%in%x,%y,%z$for%the%plane%that%passes%through%the%given%points.%
!
!P1(3, 4,%1),%P2(3,%2,%1),%P3( 1,%1, 2).
Distance from a point to a plane:
Example:
Find the distance from the point P (2, -1, 3) to the plane: 2x + 4y – z + 1 = 0