To find the area of a parallelogram when you have the coordinates of its vertices, you can use the formula based on the cross product of the vectors defined by the corners. Let’s label the vertices as follows: K(1, 2, 3), L(1, 3, 6), and M(3, 8, 6).
First, we need to determine the position of our fourth vertex, which we can denote as N. The coordinates of N can be found using vector addition. The vectors KL and KM can be calculated as:
K to L: v1 = L – K = (1, 3, 6) – (1, 2, 3) = (0, 1, 3)
K to M: v2 = M – K = (3, 8, 6) – (1, 2, 3) = (2, 6, 3)
The area A of the parallelogram is given by the magnitude of the cross product of these two vectors:
A = |v1 × v2|
Calculating the cross product:
v1 × v2 = |i j k|
|0 1 3|
|2 6 3|
To compute this determinant:
A = i(1*3 – 3*6) – j(0*3 – 3*2) + k(0*6 – 1*2)
A = i(3 – 18) – j(0 – 6) + k(0 – 2)
A = -15i + 6j – 2k
The magnitude of this vector is:
|A| = sqrt((-15)^2 + 6^2 + (-2)^2)
|A| = sqrt(225 + 36 + 4) = sqrt(265)
Therefore, the area of the parallelogram is:
A = sqrt(265)
This value gives the area enclosed by the parallelogram with the given vertices K, L, and M.