Video Transcript
If the magnitude of 𝐀 cross 𝐁
squared plus the absolute value of 𝐀 dot 𝐁 squared equals 17,424 and the magnitude
of 𝐀 is 12, find the magnitude of 𝐁.
First things first, let’s
understand why we read the vertical bars as denoting a magnitude in this term but an
absolute value in this term. This is because the cross product
of two vectors is another vector, but the dot product of two vectors is a
scalar. When we put vertical bars around a
vector, we mean find its length or magnitude. When we put vertical bars around a
scalar, we mean find its absolute value.
The reason we use the same notation
with vertical bars to represent both quantities is that the magnitude of a vector
plays a very similar role to the absolute value of a scalar. Nevertheless, because finding the
magnitude of a vector is really a different operation from finding the absolute
value of a scalar, it’s important to understand what kind of quantity is inside the
vertical bars to correctly understand what operation the vertical bars are meant to
represent.
So likewise, the vertical bars
around 𝐀 and the vertical bars around 𝐁 both represent vector magnitudes because
the quantities inside the bars are vectors. Anyway, turning back to our goal,
it’s not obvious from this equation how to come up with an expression for the
magnitude of 𝐁. However, if we focus on the
quantities inside the vertical bars, we see that we have a vector cross product and
a vector dot product, both of which can be expressed in terms of the magnitudes of
the two vectors. The dot product of two vectors 𝐀
and 𝐁 has a value given by the magnitude of 𝐀 times the magnitude of 𝐁 times the
cosine of the angle between them represented with the Greek letter 𝜃.
Note that the magnitude of 𝐀, the
magnitude of 𝐁 , and the cos of 𝜃 are all scalars. So the right-hand side of this
expression is a product of three scalar quantities, which means it itself is a
scalar, which is good because we know that 𝐀 dot 𝐁 should be a scalar.
There is a similar formula for 𝐀
cross 𝐁 in terms of the magnitude of 𝐀, the magnitude of 𝐁, and the angle between
them. But remember, 𝐀 cross 𝐁 is a
vector, so this formula will actually give us the magnitude of 𝐀 cross 𝐁. The magnitude of the vector 𝐀
cross 𝐁 is equal to the magnitude of 𝐀 times the magnitude of 𝐁 times the sine of
the angle between them. Now, since 𝐀 cross 𝐁 is a vector,
it also has a direction. And it turns out that this
direction is actually given by what’s known as the right-hand rule. But in our particular statement, we
only care about the magnitude of 𝐀 cross 𝐁, so we don’t need to worry about its
direction. Also, just like the dot product, on
the right-hand side, we have a product of three scalars, which gives us another
scalar, which is exactly what we should have because the magnitude of a vector is in
fact a scalar.
Finally, we also note that the
angle between two vectors, 𝜃, is between zero and 180 degrees. The sine of any angle between zero
and 180 degrees is positive or zero. Also, vector magnitudes are always
positive or zero. So this expression is the product
of three terms that are positive or zero, so it itself is positive or zero. And this is consistent because this
formula gives us the magnitude of a vector, which, as we just said, is positive or
zero. The dot product, however, can be
positive, zero, or negative because the magnitudes of 𝐀 and 𝐁 are always
positive. But the cosine of angles between
zero and 180 degrees can be positive, negative, or zero. So in our equation, we take the
absolute value of 𝐀 dot 𝐁 because 𝐀 dot 𝐁 can be positive or negative.
Okay, so now that we’ve expressed
both the dot product and the magnitude of the cross product in terms of the
magnitudes of 𝐀 and 𝐁, let’s plug these formulas into our equation. So we have that the magnitude of 𝐀
times the magnitude of 𝐁 times the sin of 𝜃 all squared plus the absolute value of
the magnitude of 𝐀 times the magnitude of 𝐁 times the cos of 𝜃 squared is equal
to 17,424. Note that the vertical bars around
this first term disappeared because what we replaced was the quantity the magnitude
of 𝐀 cross 𝐁. However, we still have the absolute
value bars in the second term because what we replaced was 𝐀 dot 𝐁 not the
absolute value of 𝐀 dot 𝐁.
However, 𝐀 dot 𝐁 is a real
number. And for any real number, its square
is the same as the square of its absolute value. For example, negative two squared
is negative two times negative two, which is positive four, which is the same thing
as two squared, and two is the absolute value of negative two. So, going back to our equation, we
can remove the vertical bars from around this term because the square of a real
number is always positive or zero. We are actually quite close to our
solution. The next step is to replace these
two squares of products with the products of squares. That is, the magnitude of 𝐀 times
the magnitude of 𝐁 times the sin of 𝜃 all squared is the magnitude of 𝐀 squared
times the magnitude of 𝐁 squared times the sin of 𝜃 squared.
Here, we use the convention where
we represent the sin of 𝜃 squared and the cos of 𝜃 squared with the two coming
before the 𝜃. This way, we don’t accidentally get
confused and square the angle before applying the trigonometric function. Now, observe that we have a sin 𝜃
squared and a cos 𝜃 squared in two separate terms. And both of these terms have the
same coefficient, the magnitude of 𝐀 squared times the magnitude of 𝐁 squared. This should remind us of the very
important identity that for any angle 𝜃 the cos of 𝜃 squared plus the sin of 𝜃
squared is always equal to one. If we factor out the common factor
from these two terms, we get the magnitude of 𝐀 squared times the magnitude of 𝐁
squared times the quantity cos 𝜃 squared plus sin 𝜃 squared.
But according to our trigonometric
identity, this term in the parentheses is exactly one. So we have the magnitude of 𝐀
squared times the magnitude of 𝐁 squared times one, which is just equal to the
magnitude of 𝐀 squared times the magnitude of 𝐁 squared. Now, since this is equivalent to
the left-hand side of our previous equation, it must be equal to the right-hand side
of our previous equation 17,424. And now we have exactly what we
need. The magnitude of 𝐀 squared times
the magnitude of 𝐁 squared is 17,424, which is a number on one side of an equation,
and the magnitude of 𝐀 and the magnitude of 𝐁 on the other side of the
equation. But the magnitude of 𝐁 is what
we’re looking for, and we’re given a value for the magnitude of 𝐀. So there’s actually only one
unknown quantity in this equation. And we can directly solve for the
magnitude of 𝐁.
We could start by dividing both
sides by 𝐀 squared or taking the square root of both sides. But let’s instead start by
substituting in 12 for the magnitude of 𝐀. If the magnitude of 𝐀 is 12, then
the magnitude of 𝐀 squared is 12 squared, which is 144. So 144 times the magnitude of 𝐁
squared is 17,424. And dividing both sides by 144, we
find that the magnitude of 𝐁 squared is 121. Now, 121, we may recognize, is
exactly 11 squared. If we didn’t recognize this, we
could just go ahead and take the square root of 121 and find that it is 11. Now, as usual, we have to be
careful when taking the square root of 121 because 121 has two square roots:
positive 11 and negative 11. However, recall that we are solving
for the magnitude of a vector, and the magnitude of a vector is always positive or
zero. So we want the positive square root
of 121, which is 11.
