Answer:
Area = 71.3 sq unit
Step-by-step explanation:
a) We will use properties of parallelogram to verify the given vertices.
Property: Have a pair of parallel opposite sides
Hence, vectors AB AD BD BC CD AC. A pair should be parallel:
vectors
AB = A - B = (1, 1 , 3) - ( 2 , -5 , 6 ) = (-1 , 6 , -3)
AD = A - D = (1, 1 , 3) - (3, 4 , -4) = (-2 , -3 , 7)
BC = B - C = (2 , -5 , 6) - (4, -2 , -1) = (-2 ,-3 , 7)
CD = C - D = (4, -2 , -1) - (3, 4 , -4) = (1 , -6 , 3)
Hence we can see that AB //CD and AD // BC as their unit vector co-efficents are identical/scalar multiple of each other.
b)
Equation of line AB = (1,1,3) + t*(-1 , 6 , -3 )
Point E = (1-t , 1+6t , 3 -3t) ... denotes an position of point E on line AB.
Choose a point on line CD, lets suppose C = ( 4 , -2 , -1 )
Vector EC = ( 1-t , 1+6t , 3-3t ) - (4 , -2 , -1 ) = (-3 -t , 3 +6t , 4 -3t)
For EC to be perpendicular to AB then dot product of AB . EC = 0
Hence,
EC . AB = -1 * (-3-t) -2*(3+6t) -3*(4-3t) = 0
3 + t -6 -12t -12+9t = 0
-2t-3 = 0
t = -3/2
Vector EC = (-3 -t , 3 +6t , 4 -3t) = (-1.5 , -6 , 8.5)
Area = magnitude (AB) * magnitude (EC)
Area = sqrt ((-1)^2 + 6^2 + (-3)^2) * sqrt ((-1.5)^2 + 6^2 + (8.5)^2)
Area = sqrt (46) * sqrt (442) / 2
Area = 71.3 unit^2
