I have this number that I just wanted to be sure how to do. They give me a Parallelepiped with corners lettered from A to H. Then the question is:

In a certain orthonormal reference, the coordinates of points A, B, D and E are respectively

A(0,0,2)
B(3,0,2)
D(0,4,2)
E(0,0,1)

Calculate the coordinates of point C in that same reference.

I determined that the coordinates for C are (3,4,2) but I have no idea how one goes to calculate that. I'm looking for a method to resolve this number.

Thank you
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To calculate the coordinates of point C, we can use the properties of a parallelepiped. In an orthonormal reference system, a parallelepiped can be defined by three vectors representing its edges, usually denoted as a, b, and c.

In this case, we are given the coordinates of points A, B, D, and E. We can determine the vectors AB, AD, and AE by subtracting the coordinates of the initial point from the final point:

AB = B - A = (3, 0, 2) - (0, 0, 2) = (3, 0, 0)
AD = D - A = (0, 4, 2) - (0, 0, 2) = (0, 4, 0)
AE = E - A = (0, 0, 1) - (0, 0, 2) = (0, 0, -1)

Next, we can use the fact that the cross product of two vectors represents a vector perpendicular to both of them. We can calculate the cross product of AB and AD to find vector AC, which is perpendicular to both AB and AD:

AC = AB × AD

The cross product of two vectors can be calculated using the following formula:
AC = (ABy * ADz - ABz * ADy, ABz * ADx - ABx * ADz, ABx * ADy - ABy * ADx)

Substituting the values, we have:

AC = (0 * 4 - 0 * 0, 0 * 0 - 3 * 0, 3 * 0 - 0 * 4)
= (0, 0, 0)

Since AC is the zero vector, it means that vector C is collinear with vectors A, B, D, and E. This implies that point C lies on the same plane determined by A, B, D, and E.

To find the coordinates of C, we can observe that the x-coordinate of C is 3, the y-coordinate is 4, and the z-coordinate is 2 (the same as points B and D). Therefore, the coordinates of point C in the same reference system are (3, 4, 2).