Compute the area of the parallelogram formed by A(-2, 1, 4), B(0, -2, 3) C(-1, 6, 5), and D(1, 3, 4).

So for this problem, do I turn the four points into two vectors with AD and BC and then take the cross product and then find the magnitude?

Taking the cross product and magnitude is the easy part, but I'm not sure how exactly to turn the four given coordinates into vectors.

Any help is greatly appreciated!

take the cross product of any two adjacent sides.

http://onlinemschool.com/math/assistance/vector/multiply1/
this allows you to just use the points given

sketch graph first :)

A to B = +2i-3j-1k
B to D = +1i+5j+1k
D to C = -2i+3j+1k parallel to AB
C to A = -1i-5j-1k parallel to BD

so use AB cross BD for example

To compute the area of the parallelogram formed by the given points A(-2, 1, 4), B(0, -2, 3), C(-1, 6, 5), and D(1, 3, 4), you are correct that you need to find the cross product of two vectors. Here's how you can do it:

1. Start by defining the two vectors using the given points. You can choose any two points to create the vectors, but make sure they are not collinear.

Vector AD = D - A = (1 - (-2), 3 - 1, 4 - 4) = (3, 2, 0)
Vector BC = C - B = (-1 - 0, 6 - (-2), 5 - 3) = (-1, 8, 2)

2. Next, take the cross product of these two vectors by using the determinant of the matrix formed by their components. The cross product, let's call it V, is calculated as:

V = AD x BC = (2 * 2 - 0 * 8, -(3 * 2 - 0 * (-1)), 3 * 8 - 2 * (-1)) = (4, -6, 26)

3. Now that you have the cross product vector V, find its magnitude using the formula:

magnitude = sqrt(Vx^2 + Vy^2 + Vz^2) = sqrt(4^2 + (-6)^2 + 26^2) = sqrt(16 + 36 + 676) = sqrt(728) = 2√182

4. Finally, the area of the parallelogram is equal to the magnitude of the cross product vector. Therefore, the area is:

Area = 2√182

So the area of the parallelogram formed by points A, B, C, and D is 2√182.