If π΄π΅πΆπ· is a square with a side length of 81 cm, and π is a unit vector perpendicular to its plane, find π¨π© Γ π©π.

If π΄π΅πΆπ· is a square with a side length of 81 centimeters and π is a unit vector perpendicular to its plane, find the cross product of vector π¨π© and vector π©π.

We are told that π΄π΅πΆπ· is a square with a side length of 81 centimeters. We are told that π is a unit vector perpendicular to its plane. And we want to find the cross product of vectors π¨π© and π©π. The cross product of two vectors π¨ and π© is a vector perpendicular to the plane that contains π¨ and π© and whose magnitude is given by the magnitude of vector π¨ multiplied by the magnitude of vector π© multiplied by the magnitude of sin π, where π is the angle between the two vectors.

Since each side of our square has length 81 centimeters, then the magnitude of vector π¨π© is 81. Likewise, the magnitude of vector π©π is 81. Since the vectors are the sides of a square, the angle between them is 90 degrees. The cross product of vectors π¨π© and π©π is therefore equal to 81 multiplied by 81 multiplied by the sin of 90 degrees multiplied by the unit vector π. 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 π¨π© and π©π is 6,561π.

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