Video Transcript
If 𝐀 equals negative 𝐢 hat minus
𝐣 hat plus two 𝐤 hat, 𝐁 equals 𝐣 hat plus 𝐤 hat, and 𝐂 equals negative 𝐢 hat
minus 𝐤 hat, find the magnitude of 𝐀 minus 𝐁 minus 𝐂.
In this statement, 𝐀, 𝐁, and 𝐂
are all vectors as represented by the half arrow above the letter. Each of these three vectors is
represented as a sum in terms of the three perpendicular unit vectors 𝐢 hat, 𝐣
hat, and 𝐤 hat. To find the magnitude that we’re
looking for, we will first need to calculate the difference 𝐀 minus 𝐁 minus 𝐂 and
then take the magnitude of that resulting vector. Luckily, it will be quite easy to
calculate this difference because all of the vectors are already expressed in terms
of their components.
Recall that when we represent two
vectors, say 𝐕 and 𝐖, in terms of 𝐢 hat, 𝐣 hat, and 𝐤 hat, then their
difference, 𝐕 minus 𝐖, is the difference of their 𝐢 hat components times 𝐢 hat
plus the difference of the 𝐣 hat components times 𝐣 hat plus the difference of the
𝐤 hat components times 𝐤 hat. In other words, we subtract each
component separately. When there are three vectors, like
in our calculation, we do the exact same thing, but with three numbers instead of
two for each component.
Let’s start with the 𝐢 hat
component of 𝐀 minus 𝐁 minus 𝐂. The 𝐢 hat component of 𝐀 is
negative one. 𝐁 has no 𝐢 hat component, so the
𝐢 hat component of 𝐁 is zero. And the 𝐢 hat component of 𝐂 is
also negative one. So, the 𝐢 hat component of 𝐀
minus 𝐁 minus 𝐂 is negative one minus zero minus negative one. For the 𝐣 hat component of 𝐀
minus 𝐁 minus 𝐂, the 𝐣 hat component of 𝐀 is negative one, the 𝐣 hat component
of 𝐁 is one, and our expression for 𝐂 has no 𝐣 hat in it. So, the 𝐣 hat component of 𝐂 is
zero. So, the 𝐣 hat component of 𝐀
minus 𝐁 minus 𝐂 is negative one minus one minus zero. Finally, for the 𝐤 hat component
of 𝐀 minus 𝐁 minus 𝐂, the 𝐤 hat component of 𝐀 is two, the 𝐤 hat component of
𝐁 is one, and the 𝐤 hat component of 𝐂 is negative one. So, we have two minus one minus
negative one.
Now, we just need to evaluate the
expressions in each set of parentheses. For the 𝐢 hat component, negative
one minus zero is negative one, and negative one minus negative one is zero. For the 𝐣 hat component, negative
one minus one is negative two, and negative two minus zero is still negative
two. Finally, for the 𝐤 hat component,
two minus one is one, and one minus negative one is two. So, the vector 𝐀 minus 𝐁 minus 𝐂
is negative two 𝐣 hat plus two 𝐤 hat. And we’ve omitted the 𝐢 hat term
because it is just zero.
Now, we just need to find the
magnitude of this vector. For a vector expressed in terms of
𝐢 hat, 𝐣 hat, and 𝐤 hat, the magnitude is just the square root of the sum of the
squares of the coefficients of 𝐢 hat, 𝐣 hat, and 𝐤 hat. This works because 𝐢 hat, 𝐣 hat,
and 𝐤 hat are all mutually perpendicular and have magnitude one. So, the magnitude we’re looking for
is the square root of negative two squared plus two squared, where again we’ve left
off the 𝐢 hat component because it is simply zero. Negative two squared is negative
two times negative two, which is positive four, and two squared is likewise positive
four. The square root of four plus four
is the square root of eight. And since eight is two times two
squared, the square root of eight is equal to two times the square root of two.
So, the magnitude of 𝐀 minus 𝐁
minus 𝐂 is two times the square root of two.
