### Video Transcript

Find, to the nearest degree, the
measure of the angle between the planes three π₯ minus two π¦ plus three π§ is equal
to seven and two times π₯ minus one plus three times π¦ minus four plus four times
π§ plus five is equal to zero.

In this question, weβre asked to
find the measure of the angle between two given planes, and we need to find this
angle to the nearest degree. To do this, letβs start by
recalling how we find the measure of the angle between two given planes. We know if we have two planes with
normal vectors π§ sub one and π§ sub two, then the acute angle π between the two
planes will satisfy the equation cos of π is equal to the absolute value of the dot
product of vectors π§ sub one and π§ sub two divided by the magnitude of π§ sub one
multiplied by the magnitude of π§ sub two. So, we can find an equation
involving the angle between the two planes if we can find a normal vector to each
plane.

We can find the normal vector to
both of our planes by recalling the plane ππ₯ plus ππ¦ plus ππ§ is equal to π
will have a normal vector of π, π, π. In other words, the components of
the normal vector to the plane are the coefficients of the variables. We can directly find the normal
vector of the first plane by just looking at the coefficients of the variables. We get the normal vector for the
first plane π§ sub one is equal to the vector three, negative two, three.

We now want to find the normal
vector of the second plane. And we can do this by first
distributing all of our coefficients over the parentheses. However, we donβt need to do this
in full; weβre only interested in the coefficients of our variables. So, we only need to calculate these
terms. We get two π₯, three π¦, and four
π§. And then taking these coefficients
as the components of our vector gives us the normal vectors to this plane π§ sub two
is the vector two, three, four.

We could now substitute these
vectors into our equation involving the angle π. However, itβs usually easier to
determine the numerator and denominator of the right-hand side of this equation
separately. Letβs first find the dot product of
the two vectors. We want to find the dot product of
the vector three, negative two, three and the vector two, three, four. And we can do this by recalling to
find the dot product of two vectors of equal dimension, we just need to find the sum
of the products of the corresponding components. In this case, thatβs three times
two plus negative two multiplied by three plus three times four, which is equal to
12. Remember, in the right-hand side of
our equation, we want to find the absolute value of this dot product. Well, the absolute value of 12 is
just equal to 12.

We now want to evaluate the
denominator of the right-hand side of our equation. To do this, we need to find the
magnitude of our two normal vectors. Letβs start with the magnitude of
π§ sub one. Remember, the magnitude of a vector
is the square root of the sum of the squares of its components. So, the magnitude of vector π§ sub
one is the square root of three squared plus negative two squared plus three
squared, which we can then calculate is the square root of 22. We can follow the same process to
find the magnitude of vector π§ sub two. Itβs equal to the square root of
two squared plus three squared plus four squared, which we can evaluate is root
29.

Weβre now ready to substitute these
values into our equation. This gives us that the cos of π is
equal to 12 divided by root 22 times root 29. We could simplify the right-hand
side of this equation. However, itβs not necessary since
we now need to take the inverse cosine of both sides of the equation anyway. This gives us that π is equal to
the inverse cos of 12 divided by root 22 times root 29.

And if we input this expression
into our calculator, where we make sure itβs set to degrees mode, we get 61.63 and
this expansion continues degrees. But remember, the question wants us
to determine this angle to the nearest degree. So, we need to look at the first
decimal digit, which is six. This is greater than or equal to
five. So, we need to round this value up,
which then gives us our final answer.

The measure of the angle between
the planes three π₯ minus two π¦ plus three π§ is equal to seven and two times π₯
minus one plus three times π¦ minus four plus four times π§ plus five is equal to
zero to the nearest degree is 62 degrees.