Question

verify that the points are the vertices of a parallelogram, and find its area. a(1, 1, 3), b(9, −3, 0), c(11, 0, −7), d(3, 4, −4)

Answer 1

The given points are the vertices of a parallelogram, and its area is approximately 25.352 square units.

Step-by-step explanation:

To verify if the given points are the vertices of a parallelogram, we need to check if the opposite sides of the quadrilateral are parallel. Let's calculate the vectors for each pair of opposite sides:

Vector AB: (9 - 1, -3 - 1, 0 - 3) = (8, -4, -3)

Vector DC: (3 - 11, 4 - 0, -4 - (-7)) = (-8, 4, -3)

Vector BC: (11 - 9, 0 - (-3), -7 - 0) = (2, 3, -7)

Vector DA: (1 - 3, 1 - 4, 3 - (-4)) = (-2, -3, 7)

If the vectors AB and DC are parallel, and the vectors BC and DA are parallel, then the points are the vertices of a parallelogram.

Next, we can calculate the area of the parallelogram using the formula:

Area = magnitude of vector AB x magnitude of vector BC x sin(angle between AB and BC)

Let's calculate the magnitudes and the angle:

Magnitude of vector AB = √(8² + (-4)² + (-3)²) = √(64 + 16 + 9) = √89

Magnitude of vector BC = √(2² + 3² + (-7)²) = √(4 + 9 + 49) = √62

Angle between AB and BC can be calculated using the dot product: AB · BC = (8)(2) + (-4)(3) + (-3)(-7) = 16 - 12 + 21 = 25

Area = √89 x √62 x sin(25°) = √(89 x 62) x 0.4226 ≈ 25.352 square units