Question

Verify that the points are the vertices of a parallelogram, and find its area. (1,1,1),(2,3,4),(6,5,2),(7,7,5)

Short answer

The points form a parallelogram. The area of the parallelogram is \( 10\sqrt{2} \) units squared.

Step-by-step solution

Determine the Vectors

To start, determine the vectors that represent the sides of the parallelogram. This can be done by subtracting the coordinates of one vertex from the coordinates of another vertex. Let's denote the points as A(1,1,1), B(2,3,4), C(6,5,2) and D(7,7,5). So the vectors AB, BC, CD and DA can be found like:\[ \vec{AB}=B-A=(2-1,3-1,4-1)=(1,2,3) \]\[ \vec{BC}=C-B=(6-2,5-3,2-4)=(4,2,-2) \]\[ \vec{CD}=D-C=(7-6,7-5,5-2)=(1,2,3) \]\[ \vec{DA}=A-D=(1-7,1-7,1-5)=(-6,-6,-4) \]

Verify that both pairs of Opposite Sides are Equal

Using these vectors, demonstrate that the opposite sides of the parallelogram are equal. In other words, we have to show that \( \vec{AB}=\vec{CD} \) and \( \vec{BC}=\vec{DA} \). From the vectors calculated in Step 1, we can see that \(\vec{AB}=(1,2,3) = \vec{CD}=(1,2,3) \) and \( \vec{BC}=(4,2,-2) = - \vec{DA}=(-4,-2,2) \) Therefore, AB is parallel to CD and BC is parallel to DA which means they are vertices of a parallelogram.

Calculate the Area of the Parallelogram

The area of a parallelogram is given by the magnitude of the cross product of two adjacent sides. Select any two adjacent sides, say AB and BC. The cross product is given by: \[ AB \times BC = (1,2,3) \times (4,2,-2) = (2*-2-3*2 , 3*4-1*-2, 1*2-2*4) = (-10, 8, -4) \] The magnitude of this vector (which is the area) is given by: \[ |AB \times BC| = \sqrt{(-10)^2 + 8^2 + (-4)^2} = \sqrt{200}=10\sqrt{2} \]