Given that 𝐴𝐵𝐶𝐷 is a square
with side length 27 centimeters and 𝐞 hat is the unit vector perpendicular to its
plane, determine the cross product of 𝐀𝐁 and 𝐂𝐀.
In this question, we’re being asked
to calculate the cross product. And we know the cross product of
vector 𝐚 and vector 𝐛 is equal to the magnitude of vector 𝐚 multiplied by the
magnitude of vector 𝐛 multiplied by sin of angle 𝜃 multiplied by the unit vector
𝐧. 𝜃 is the angle between the two
vectors. And the unit vector 𝐧 is
perpendicular to both vector 𝐚 and vector 𝐛.
We are told that the square
𝐴𝐵𝐶𝐷 has side length 27 centimeters. As the magnitude of a vector is
equal to its length, the magnitude of vector 𝐀𝐁 is equal to 27. We also need to calculate the
magnitude of vector 𝐂𝐀. As triangle 𝐴𝐵𝐶 is a right
triangle, we can do this using the Pythagorean theorem, which states that 𝑎 squared
plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the longest side
In this question, we have the
magnitude of 𝐀𝐁 squared plus the magnitude of 𝐁𝐂 squared is equal to the
magnitude of 𝐂𝐀 squared. We know that the magnitude or
length of 𝐀𝐁 and 𝐁𝐂 is 27. 27 squared plus 27 squared is equal
to 1458. We can then square root both sides
of this equation so that the magnitude of 𝐂𝐀 is equal to 27 root two.
Our next step is to redraw our
diagram so that the tails or start points of both vectors are at the same point. We need to calculate the angle 𝜃
between vector 𝐀𝐁 and vector 𝐂𝐀. As the diagonal in a square cuts a
right angle in half and a half of 90 degrees is 45 degrees, then 𝜃 will be equal to
180 minus 45 degrees. The angle between vector 𝐀𝐁 and
vector 𝐂𝐀 is 135 degrees.
We can now calculate the cross
product of vector 𝐀𝐁 and vector 𝐂𝐀. 𝐀𝐁 cross 𝐂𝐀 is equal to the
magnitude of 𝐀𝐁 multiplied by the magnitude of 𝐂𝐀 multiplied by sin of angle 𝜃
multiplied by the unit vector 𝐞, as this is the unit vector perpendicular to the
plane. Substituting in our values, we have
27 multiplied by 27 root two multiplied by sin of 135 degrees multiplied by the unit
vector 𝐞. sin of 135 degrees is equal to root
two over two. We need to multiply this by 27, 27
root two, and the unit vector 𝐞. Root two multiplied by root two
over two is equal to one. So we are left with 27 multiplied
by 27 multiplied by the unit vector 𝐞. This is equal to 729𝐞. The cross product of 𝐀𝐁 and 𝐂𝐀
is equal to 729𝐞.