Question

Find the area of the parallelogram with one corner at P1 and adjacent sides P1P2 and P1P3. NOTE: There is an arrow over P1P2 and P1P3. What does that arrow mean?

P1 = (0, 0, 0), P2 = (2, 3, 1),
P3 = (-2, 4, 1)

Answer 1

I think you need to get the height * width.

Answer 2

I don't understand why you are using three coordinates for the parallelogram corner points instead of two. Is the parallelogram a tilted plane figure in three dimensions, not located on the x,y plane? Are we supposed to determine the location of the fourth corner?

Answer 3

If this parallelogram really is located in three dimensions, then a quick way to get the area is to take the cross product of the vector from P1 to P2 and the vector from P1 to P3. I think that's what they want you to do. The magnitude of the cross product vector is the area.

Answer 4

P1P2 with an arrow over it is the vector representation of the side that goes from P1 to P2. It would be written
2i + 3j + k
P1P3(with arrow), the adjacent side, would be written
-2 +4j + k
The vector whose magnitude is the parallelogram area is
|i j k|
|2 3 1| = -i +0j +2k
|-2 4 1|

(The 3x3 matrix above, surrounded by | | indicates a determinant).
The area is the magnitude of that vector, which is sqrt5 = 2.236