Find the general form of the
equation of the plane which intersects the coordinate axes at the points two, zero,
zero; zero, eight, zero; and zero, zero, four.
Okay, so in this example, we’re
told that we have a plane that intersects the 𝑥-, 𝑦-, and 𝑧-axes at certain
points. On our sketch, if we say that those
points are, say, here, here, and here, then we can write in the coordinates of those
points. The intersection point on the
𝑥-axis is at two, zero, zero. That on the 𝑦-axis is at zero,
eight, zero, and on the 𝑧-axis, zero, zero, four.
When we know the intersection
points of a plane and the coordinate axes, then we can write the equation of that
plane in what’s called intercept form. In this way of writing a plane’s
equation, these denominator values, 𝐴, 𝐵, and 𝐶, correspond to the points of
intersection of our plane and the coordinate frame. That is, 𝐴 is equal to the
𝑥-coordinate of the point of intersection of the plane and the 𝑥-axis. Likewise, 𝐵 is equal to the
𝑦-coordinate of the point of intersection of the plane and the 𝑦-axis. And a similar relation holds true
for 𝐶; it’s equal to the 𝑧-coordinate of the 𝑧-axis intersection point.
Since we’re given these values,
two, eight, and four, respectively, we can write the equation of our plane in
intercept form. That is, 𝑥 divided by two plus 𝑦
divided by eight plus 𝑧 divided by four equals one. We remember though that, really, we
want to write the equation of our plane in general form. Expressed that way, we write it as
𝑎 times 𝑥 plus 𝑏 times 𝑦 plus 𝑐 times 𝑧 plus 𝑑 equals zero.
To start converting this form of
our plane, intercept form, into general form, we can take a first step by
multiplying both sides of the equation by the product of all the denominators on our
left-hand side. That is, we multiply both sides by
two times four times eight. We take this step so that when we
distribute these values across the terms in parentheses, all the denominators will
turn to one. Note that, alternatively, we
could’ve multiplied both sides simply by eight. Let’s see how this works though
term by term.
If we multiply 𝑥 divided by two by
two times four times eight, then the twos cancel out. And four times eight being 32, we
get 32𝑥. For our next term, if we multiply
𝑦 divided by eight by two times four times eight, this time the eights cancel
out. And we’re left with two times four,
or eight, times 𝑦. Lastly, 𝑧 over four times two
times four times eight gives us two times eight or 16𝑧. And all of this is equal on the
right-hand side to two times four times eight.
Going one step further, notice that
everything on the left and the right is evenly divisible by eight. We get then four 𝑥 plus 𝑦 plus
two 𝑧 being equal to two times four, or eight. And if we then subtract eight from
both sides, we now have the equation of our plane written in general form. In general form then, our plane has
the equation four 𝑥 plus 𝑦 plus two 𝑧 minus eight equals zero.