a) To determine the area of triangle ABC, we can use the formula for the magnitude of the cross product of two vectors.
First, we need to find two vectors that are in the plane of triangle ABC. Let's choose vectors AB and AC.
Vector AB can be found by subtracting the coordinates of point A from the coordinates of point B:
AB = B - A = (4, -1, 7) - (3, 2, -5) = (1, -3, 12)
Vector AC can be found by subtracting the coordinates of point A from the coordinates of point C:
AC = C - A = (-8, 3, -6) - (3, 2, -5) = (-11, 1, -1)
Next, we can calculate the cross product of vectors AB and AC:
AB x AC = (1, -3, 12) x (-11, 1, -1) = [(12)(-1) - (-3)(-1), (1)(-1) - (12)(-11), (1)(-3) - (1)(-11)]
= (-9, -133, 8)
The magnitude of this cross product gives us the area of triangle ABC:
Area = ||AB x AC|| = √((-9)^2 + (-133)^2 + 8^2) = √(81 + 17689 + 64) = √17834 ≈ 133.60
Therefore, the area of triangle ABC is approximately 133.60 square units.
b) To determine the coordinates of point D such that ABCD is a parallelogram, we need to find a vector that is parallel to AB and has the same magnitude as AC.
One way to accomplish this is to add vector AC to the coordinates of point B:
D = B + AC = (4, -1, 7) + (-11, 1, -1) = (-7, 0, 6)
Therefore, the coordinates of point D are (-7, 0, 6) such that ABCD is a parallelogram.
c) To determine if the parallelogram ABCD is a rectangle, we need to check if the diagonals are perpendicular to each other.
The diagonals of ABCD are AD and BC. Let's find the vectors for these diagonals:
Vector AD = D - A = (-7, 0, 6) - (3, 2, -5) = (-10, -2, 11)
Vector BC = C - B = (-8, 3, -6) - (4, -1, 7) = (-12, 4, -13)
Now, we can calculate the dot product of vectors AD and BC. If the dot product is 0, then the diagonals are perpendicular.
AD · BC = (-10)(-12) + (-2)(4) + (11)(-13) = 120 - 8 - 143 = -31
Since the dot product is not 0, the diagonals AD and BC are not perpendicular. Therefore, the parallelogram ABCD is not a rectangle.
And that's how we determine the area of triangle ABC, find the coordinates of point D to form a parallelogram, and check if the parallelogram is a rectangle.