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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