Area of Parallelogram With Vectors?

Question

I'm having some trouble setting up the vectors needed to solve for the area of a parallelogram

The vertices are K(1,2,3) L(1,3,6) M(3,8,6) N(3,7,3)

I should be able to reduce these down to a 2x2 matrix, correct?
If so, which vectors do I need to use? Would it be something like vector LM and vector KN?

Thanks

Answer 1

First let's verify that this is a parallelogram. Let's check the vectors of opposite sides.

KL = <0, 1, 3>
NM = <0, 1, 3>

LM = <2, 5, 0>
KN = <2, 5, 0>

So it is a parallelogram. Now calculate the area. Take the magnitude of the cross product of adjacent vectors.

Area = | KL X KN | = |<0, 1, 3> X <2, 5, 0>|
Area = |<-15, 6, -2>| = √[(-15)² + 6² + (-2)²]
Area = √(225 + 36 + 4) = √265 ≈ 16.278821

Answer 2

The area = || A X B || ( X means cross product ) = 5/2* 1/2 -(-3)*1 = 5/4 + 3 = 4.25 square units

Answer 3

You can compute the magnitude of the cross product of two vectors from the same vertex going to adjacent vertices of that vertex. For example you can do KL x KN where the two vectors start from K and go to the vertices adjacent to K, L and N.