Can someone please help me understand the difference between the following:
- Find the equation of the plane that passes through the point $(2,-1,3)$ and is parallel to the plane $5x-2y+z-5=0$.
In this case, is the plane's normal vector equal to the other plane's $(5x-2y+z-5=0)$ normal vector? i.e. $n=(5,-2,1)$ Since they are parallel?
∴ $(x-2,y+1,z-3).(5,-2,1)=0$, would then be the equation in vector form?
- Find the equation of the plane that passes through the point $(1,-1,2)$ and is orthogonal/perpendicular to the plane $x+2y-3z-4=0$
In this case, where the two planes are orthogonal, is the plane's normal vector also equal to the other plane's $(x+2y-3z-4=0)$ normal vector? i.e. $n=(1,2,-3)$
∴ $(x-1,y+1,z-2).(1,2,-3)=0$, would then be the equation in vector form?
- Find the equation of the plane that passes through the point $(1,-3,-4)$ and is orthogonal/perpendicular to the line: $x=4+2t, y=-2+3t, z=-5t$
In this case, where the plane is now orthogonal to a line, is the plane's normal vector equal to the line's $(x=4+2t, y=-2+3t, z=-5t)$ vector? i.e. $n=v=(2,3,-5)$
∴ $(x-1,y+3,z+4).(2,3,-5)=0$, would then be the plane's equation in vector form?