Find the area of the parallelogram with vertices 𝐴(−1,2,4) A ( − 1 , 2 , 4 ) , 𝐵(0,4,6) B ( 0 , 4 , 6 ) , 𝐶(1,1,5) C ( 1 , 1 , 5 ) , and 𝐷(2,3,7) D ( 2 , 3 , 7 ) .

Question

Find the area of the parallelogram with vertices , , , and .

Answer 1

translate so A is (0,0,0)

makes B (1,2,2) C (2,-1,1) D (3,1,3)

see AD=AB+AC

so vectors AB and AC define parallelogram

area is cross poduct magnitude |ABxAC|

recall |ABxAC| = |AB||AC|sinθ, where θ is angle between AB and AC and 0<θ<π

call| AB| base, then |AC|sinθ is height ,,, base times height=area

the expression for the vector ABxAC concludes

ABxAC= 4i+3j-5k

|ABxAC|=5√2