Find the vector form of the
equation of the plane that has normal vector 𝐧 equals 𝐢 hat plus 𝐣 hat plus 𝐤
hat and contains the point two, six, six.
Okay, so in this example, we have a
plane. We’ll say this is it. And we’re told that relative to
some set of coordinate axes, there’s a vector normal or perpendicular to the
plane. And if we write that vector in
vector form, we see that it has components of one, one, and one in the 𝑥-, 𝑦-, and
𝑧-directions, respectively. Along with this, we’re told that
the plane contains a point, we’ll call it 𝑃 zero, with coordinates two, six,
six. And knowing all this, we want to
solve for the vector form of the plane’s equation.
To write the plane’s equation that
way, we’ll want to define a vector that lies in the plane so that if we take the dot
product of that vector and the normal vector 𝐧, we get zero. Here’s how we can go about doing
that. First, let’s define a vector from
our origin to the point 𝑃 zero. We’ll call the vector 𝐫 zero. And since it comes from the origin,
it must have components two, six, six. And next, let’s do this. Let’s pick a point at random in our
plane, we’ll call it 𝑃, and we’ll say this point has coordinates 𝑥, 𝑦, 𝑧.
We don’t specify what these values
are, but nonetheless this point will help us because now we can draw a vector from
our origin to this point 𝑃, call that vector 𝐫, which we see has components 𝑥,
𝑦, 𝑧. And then if we subtract the vector
𝐫 zero from 𝐫, and this was the whole purpose of defining 𝐫 in the first place,
then we get a vector shown here, which lies in the plane, which means that this
vector is indeed perpendicular to the normal vector 𝐧. And that means if we take the dot
product of 𝐧 and 𝐫 minus 𝐫 zero, the result we’ll get is zero.
Another way to write this is 𝐧 dot
𝐫 is equal to 𝐧 dot 𝐫 zero. And we can now substitute in the
known values for the normal vector 𝐧 and the vector 𝐫 zero. We’ll do that down here, where we
see that the vector one, one, one dotted with 𝐫 is equal to one, one, one dotted
with two, six, six. We see the vector 𝐫 is made up of
the components 𝑥, 𝑦, 𝑧. But since we don’t know those more
specifically, we’ll leave this left-hand side as it is. On the right-hand side, though, we
can compute this dot product by multiplying the respective components together,
giving us two plus six plus six.
And since these numbers add up to
14, we can write a simplified form of our equation like this. This is the vector form of the
equation of our plane.