Video Transcript
Find the equation of the plane
passing through the point 𝑎, 𝑏, 𝑐 and parallel to the plane 𝑥 plus 𝑦 plus 𝑧
equals zero.
All right, so we want to solve for
the equation of a plane, and we know that this plane passes through a point with
coordinates 𝑎, 𝑏, 𝑐. So if we draw a sketch, let’s say
that this is the plane whose equation we want to solve for and we know that
somewhere on this plane is this point, 𝑎, 𝑏, 𝑐. Along with this, we’re told that
this plane is parallel to another plane whose equation is given here. And note that this plane’s equation
is given in such a form that we’re able to use it to solve for the components of the
plane’s normal vector. Note that all three of the
variables 𝑥, 𝑦, and 𝑧 are effectively being multiplied by one. These multiplying factors, these
ones, give us components of a vector that’s normal to this plane.
If we call that vector 𝐧, its
𝑥-component could be given by one, its 𝑦-component by one, and its 𝑧-component by
the same value. And it’s helpful to know this
normal vector because we’re told that the plane whose equation we want to solve for
is parallel to the plane whose equation is given here. And therefore, the normal vector of
this unknown plane, we’ll call this vector 𝐧 sub 𝑃, must be parallel or
antiparallel to 𝐧. And in fact, we could set 𝐧 sub 𝑃
equal to 𝐧 in general. And we do this based on our
knowledge of the fact that the two planes we’re considering are parallel.
At this point, note that we have a
vector that’s normal to the plane whose equation we want to solve for. And we also have the coordinates of
a point on that plane. Taken together, these two pieces of
information are enough to let us solve for this plane’s equation. We can recall that given a normal
vector to a plane as well as a vector to a random point on that plane, the dot
product of those two vectors equals the dot product of the normal vector with a
vector to a known point on the plane. To apply this relationship to our
scenario, we’ll say that our normal vector 𝐧 sub 𝑃 dotted with a vector that goes
to any point on our plane with the general components 𝑥, 𝑦, and 𝑧 is equal to our
normal vector 𝐧 sub 𝑃 dotted with the vector to our known point 𝑎, 𝑏, 𝑐.
We can now substitute in the value
for our normal vector 𝐧 sub 𝑃. That gives us this equation
here. And if we then carry out both of
these dot product operations, we find that 𝑥 plus 𝑦 plus 𝑧 is equal to 𝑎 plus 𝑏
plus 𝑐. And this result is the equation of
the plane that passes through the point 𝑎, 𝑏, 𝑐 and is parallel to the plane 𝑥
plus 𝑦 plus 𝑧 equals zero.
