### Video Transcript

If 𝐴𝐵𝐶 is a triangle of area
248.5 square centimeters, find the value of the magnitude of the cross product of
vectors 𝚩𝚨 and 𝚨𝐂.

We will begin by sketching triangle
𝐴𝐵𝐶. We are told that it has an area of
248.5 square centimeters. And we recall that we can calculate
the area of any triangle using the formula a half 𝑏𝑐 multiplied by sin 𝐴. In this question, we are asked to
find the value of the magnitude of the cross product of vectors 𝚩𝚨 and 𝚨𝐂. We know that the magnitude of the
cross product of two vectors 𝚨 and 𝚩 is equal to the magnitude of vector 𝚨
multiplied by the magnitude of vector 𝚩 multiplied by the magnitude of sin 𝜃.

In this question, the magnitude of
the cross product of vectors 𝚩𝚨 and 𝚨𝐂 is the magnitude of vector 𝚩𝚨
multiplied by the magnitude of 𝚨𝐂 multiplied by the magnitude of sin 𝜃, where 𝜃
is the angle between the two vectors. Extending the line 𝐵𝐴, we see
that the angle 𝜃 between vectors 𝚩𝚨 and 𝚨𝐂 is as shown. This means that the measure of
angle 𝜃 and the measure of angle 𝐴 sum to 180 degrees. And the measure of angle 𝜃 is
therefore equal to 180 degrees minus the measure of angle 𝐴.

The magnitude of vector 𝚩𝚨 is
equal to the length of the side 𝐵𝐴 in our triangle. And we have labeled this lowercase
𝑐 as it is opposite angle 𝐶 in the triangle. Likewise, the magnitude of vector
𝚨𝐂 is equal to the length of side 𝐴𝐶, and this is equal to lowercase 𝑏. The cross product is therefore
equal to 𝑐 multiplied by 𝑏 multiplied by the magnitude of sin of 180 degrees minus
𝐴. Since the sin of 180 degrees minus
𝜃 is equal to sin 𝜃, our expression is equal to 𝑐 multiplied by 𝑏 multiplied by
sin 𝐴, which can be rewritten as 𝑏𝑐 multiplied by sin 𝐴.

This is very similar to the formula
for the area of a triangle. Since the area of our triangle is
248.5 square centimeters, then 248.5 is equal to a half 𝑏𝑐 multiplied by sin
𝐴. Multiplying both sides of this
equation by two, we have 497 is equal to 𝑏𝑐 multiplied by sin 𝐴. We now have the exact same
expression as in the cross product. And we can therefore conclude that
the magnitude of the cross product of vectors 𝚩𝚨 and 𝚨𝐂 is 497.

This leads us to a general formula
for the area of a triangle. One-half of the magnitude of the
cross product of vectors 𝚨 and 𝚩 is the area of a triangle that is spanned by
vectors 𝚨 and 𝚩.