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The planes are perpendicular if the normal vectors of the planes are perpendicular.
If the dot product equals zero ,the planes are perpendicular.
x+y+z=0 the normal vector n1= <1,1,1>
x-2y+z=0 the normal vector n2=<1,-2,1>
The dot product is (1*1)+(1*-2)+(1*1)= 0
then the planes are perpendicular.
If ax+by +cz = d is a plane , then a, b and c are the direction ratios of the plane.
On the otherhand the direcction ratio of the line joining the origin and the feet of perpendicuar of the plane is also a, b,c.
So if a straight given straight line is perpendicular to a given plane then it has to be parallel to the line from the origin O to the the feet of perpendicular on the plane, Thus if the direction ratios of a line are a1, b1 ans c1 (and that of the plane are a,b,c), then
There are several methods in order to prove that a line is perpendicular to a plane (P):
-there is a line d1 perpendicular to a line d, where d belongs to (P) => d1 is perpendicular to (P), too.
- d parallel to another line d1, where d1 is perpendicular to (P), so d is perpendicular to (P), too.
- d is perpendicular to a plane (Q), where (Q) is also parallel to (P), so d is perpendicular to (P), too.
- there are 2 concurrent lines, a and b, which belong to (P), d is perpendicular to a and d is perpendicular to b, so d is perpendicular to (P), too.
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