Finding the Area of a Parallelogram in 3D Space

To find the area of a parallelogram in 3D space, we can use the formula:

Area = |u × v|

where u and v are the vectors representing two adjacent sides of the parallelogram.

Given the vertices of the parallelogram as:

• K(1, 3, 3)
• L(1, 4, 4)
• M(4, 8, 4)
• N(4, 7, 3)

Let's start by finding the vectors representing two adjacent sides of the parallelogram.

1. Vector u = L - K = (1, 4, 4) - (1, 3, 3) = (0, 1, 1)
2. Vector v = M - L = (4, 8, 4) - (1, 4, 4) = (3, 4, 0)

Now, we can calculate the cross product of these two vectors to find the area of the parallelogram:

u × v = (i, j, k) where:

• i = u₂v₃ - u₃v₂ = (1)(0) - (1)(4) = -4
• j = u₃v₁ - u₁v₃ = (1)(3) - (0)(4) = 3
• k = u₁v₂ - u₂v₁ = (0)(4) - (1)(3) = -3

Therefore, the area of the parallelogram is:

Area = |u × v| = |(i, j, k)| = |(−4, 3, −3)| = √((-4)^2 + (3)^2 + (-3)^2) = √(16 + 9 + 9) = √34

So, the area of the parallelogram with the given vertices is √34 square units.