### Video Transcript

Given that the point seven minus
π, 12 minus π, π lies in the π¦π§-plane, what is this point?

In this question, weβre given the
coordinates of a point in three dimensions, and we can see this contains an unknown
value of π. We need to determine these
coordinates, which means we need to determine the value of π. And to do this, weβre told that
this point lies in the π¦π§-plane. So to answer this question, weβre
first going to need to recall what it means for a point to be in the π¦π§-plane. If a point lies in the π¦π§-plane,
then its π¦- and π§-coordinates can take any values. However, its π₯-coordinate has to
be equal to zero. So, in particular, any point on the
π¦π§-plane will have the form zero, π, π, where π and π can be any real
numbers.

Therefore, because the point given
to us in the question lies in the π¦π§-plane, we can conclude its π₯-coordinate must
be equal to zero. So setting this equal to zero, we
get the equation seven minus π is equal to zero. And we can solve for the value of
π. Weβll add π to both sides of our
equation, giving us that our value of π is equal to seven. Now, all we need to do is
substitute π is equal to seven into our point. Substituting π is equal to seven
into our point, we get the point seven minus seven, 12 minus seven, seven. And if we calculate each of these
expressions, we get the point zero, five, seven which is our final answer.

Therefore, given that the point
seven minus π, 12 minus π, π lies in the π¦π§-plane, we were able to determine
that its π₯-coordinate must be equal to zero. This let us find the value of the
unknown π to be equal to seven. And we could use this to find the
coordinates of our point. These coordinates were zero, five,
seven.