Video Transcript
If 𝐴𝐵𝐶𝐷 is a square with a side
length of 81 centimeters and 𝐞 is a unit vector perpendicular to its plane, find
the cross product of vector 𝚨𝚩 and vector 𝚩𝐂.
We are told that 𝐴𝐵𝐶𝐷 is a
square with a side length of 81 centimeters. We are told that 𝐞 is a unit
vector perpendicular to its plane. And we want to find the cross
product of vectors 𝚨𝚩 and 𝚩𝐂. The cross product of two vectors 𝚨
and 𝚩 is a vector perpendicular to the plane that contains 𝚨 and 𝚩 and whose
magnitude is given by the magnitude of vector 𝚨 multiplied by the magnitude of
vector 𝚩 multiplied by the magnitude of sin 𝜃, where 𝜃 is the angle between the
two vectors.
Since each side of our square has
length 81 centimeters, then the magnitude of vector 𝚨𝚩 is 81. Likewise, the magnitude of vector
𝚩𝐂 is 81. Since the vectors are the sides of
a square, the angle between them is 90 degrees. The cross product of vectors 𝚨𝚩
and 𝚩𝐂 is therefore equal to 81 multiplied by 81 multiplied by the sin of 90
degrees multiplied by the unit vector 𝐞. We know that the sin of 90 degrees
is equal to one. 81 multiplied by 81 is 6,561, which
means that the cross product of 𝚨𝚩 and 𝚩𝐂 is 6,561𝐞.
