### Video Transcript

Find, to the nearest second, the
measure of the angle between the planes negative nine 𝑥 minus six 𝑦 plus five 𝑧
equals negative eight and two 𝑥 plus two 𝑦 plus seven 𝑧 equals negative
eight.

Okay, so here we have these
equations describing two different planes, and we see that they’re almost but not
quite written in general form. We recall that in general form, the
equation of a plane has zero on one side. For both of these expressions
though, if we add positive eight to both sides, then we’ll get these equations in
general form. That’s useful to us because now we
can more easily pick out the components of vectors that are normal to each one of
these planes.

Let’s say that the plane
represented by this first equation is plane one. And we’ll call the plane
represented by the second equation plane two. When a plane’s equation is given in
general form, that means whatever factors we use to multiply 𝑥, 𝑦, and 𝑧 are the
components of a vector that is normal to this plane. That is, if we call 𝐧 one a vector
normal to plane one, then we know there exists such a vector with components
negative nine, negative six, five. Similarly, for plane two, where we
can define a normal vector 𝐧 two, this vector will have components two, two,
seven. We’ve gone to the effort of solving
for vectors normal to each one of our planes because knowing these components, we’re
fairly close to solving for the measure of the angle between the planes.

If we call the angle between two
planes in general 𝛼, then the cos of 𝛼 is equal to the magnitude of the dot
product of vectors normal to each plane divided by the product of the magnitudes of
each of these vectors. Knowing 𝐧 one and 𝐧 two for our
two planes, we can use this relationship to solve for 𝛼. If we focus first on calculating
the magnitude of the dot product of 𝐧 one and 𝐧 two, that equals the magnitude of
the dot product of these two vectors. And carrying out this dot product
by multiplying together the respective components, we get negative 18 minus 12 plus
35. That’s equal to the magnitude of
five or simply five.

Next, we can calculate the
denominator of this fraction, the product of the magnitudes of our two normal
vectors. That’s equal to the square root of
negative nine squared plus negative six squared plus five squared multiplied by the
square root of two squared plus two squared plus seven squared. This equals the square root of 81
plus 36 plus 25 times the square root of four plus four plus 49 or the square root
of 142 times the square root of 57. This is equal to the square root of
142 times 57 or 8094.

Now that we’ve calculated our
numerator and denominator, we can write that the cosine of the angle between our two
planes is equal to five divided by the square root of 8094. This means that the angle 𝛼 is the
inverse cos of five over the square root of 8094. And if we enter this expression on
our calculator and round the result to the nearest second, our result is 86 degrees,
48 minutes, and 51 seconds. This is the angle in degrees,
minutes, and seconds between our two planes.