### Video Transcript

Some vectors that represent forces
are drawn to scale on a square grid. The sides of the squares are one
centimeter long. A distance of one centimeter on the
grid represents one newton of force. The difference between the
magnitude of the larger horizontal force and the magnitude of the smaller horizontal
force is Ξπ
one. The difference between the
magnitude of the larger vertical force and the magnitude of the smaller horizontal
force is Ξπ
two. What is the absolute value of Ξπ
one minus the absolute value of Ξπ
two?

The question asks us about the
following quantity: the absolute value of Ξπ
one minus the absolute value of Ξπ
two. But what does the absolute value of
Ξπ
one minus the absolute value of Ξπ
two represent? Letβs note that Ξπ
one and Ξπ
two are both differences of magnitudes of vectors. This means that Ξπ
one and Ξπ
two are scalars; they are not vectors. So, the absolute value of Ξπ
one
minus the absolute value of Ξπ
two is the difference of the absolute value of two
scalars, not the result of adding two vectors.

That said, letβs continue with our
problem. The question tells us that the side
of a square measures one centimeter and that the distance of one centimeter
represents a newton of force. Now we count the squares that
measure each vector. The red force vector has a length
of 12 spaces and therefore of 12 centimeters. It then represents a magnitude of
12 newtons. We will say that π
sub π
equals
12 newtons, which is the magnitude of the red vector. The vector is horizontal and points
to the left.

The other horizontal vector is the
blue vector. This vector points to the
right. Now we count how many squares its
length is. There are 10 squares, meaning 10
centimeters. Therefore, the magnitude of the
blue vector is 10 newtons. So, π
sub π΅ equals 10
newtons. Now we have the value of Ξπ
one,
which is the difference between the magnitude of the horizontal longest force vector
and the magnitude of the horizontal shortest vector. We see then that Ξπ
one equals 12
newtons minus 10 newtons, which equals two newtons.

The question also asks for the
difference between the magnitude of the larger vertical force and the magnitude of
the smaller horizontal force. If we now consider vertical
vectors, we can see that for the longest vertical vector of green color, π
sub πΊ
equals eight newtons because it measures eight centimeters. And for the shortest vertical
vector of purple color, π
sub π equals five newtons because it measures five
centimeters.

The magnitude of the longest
vertical vector is required. This magnitude is compared to the
magnitude of the smaller horizontal vector. We have already seen that the
smaller horizontal vector is π
sub π΅, which has a magnitude of 10 newtons. So, Ξπ
two equals 10 newtons
minus eight newtons, which equals two newtons.

We can now calculate the absolute
value of Ξπ
one minus the absolute value of Ξπ
two. This will equal the absolute value
of two newtons minus the absolute value of two newtons, which equals zero
newtons. And so, the absolute value of Ξπ
one minus the absolute value of Ξπ
two is zero newtons.