Three verticies of a parallelogram are D(2,1,3), E(-4,2,0) and F(6.-2,4). Find all possible locations of the fourth vertex.
To find all possible locations of the fourth vertex of the parallelogram, we need to understand the properties of a parallelogram.
In a parallelogram, the opposite sides are parallel and equal in length. This means that the vector connecting D to E (DE) is equal and parallel to the vector connecting F to the fourth vertex (FV). Similarly, the vector connecting D to F (DF) is equal and parallel to the vector connecting E to the fourth vertex (EV).
Using the given points D, E, and F, we can find the vectors DE and DF:
Vector DE = E - D = (-4, 2, 0) - (2, 1, 3) = (-6, 1, -3)
Vector DF = F - D = (6, -2, 4) - (2, 1, 3) = (4, -3, 1)
Now we can express the vectors connecting the fourth vertex (V) to E and F:
Vector EV = DE = (-6, 1, -3)
Vector FV = DF = (4, -3, 1)
Let's assume the coordinates of the fourth vertex are (x, y, z). We can set up equations using the vector properties mentioned above:
(x, y, z) - E = (-6, 1, -3)
(x, y, z) - F = (4, -3, 1)
Solving these two equations will give us the coordinates of the fourth vertex.
Equation 1: (x, y, z) - E = (-6, 1, -3)
Expanding the equation, we get:
x - (-4) = -6
y - 2 = 1
z - 0 = -3
Simplifying the equations, we get:
x + 4 = -6 => x = -10
y - 2 = 1 => y = 3
z - 0 = -3 => z = -3
So, one possible location of the fourth vertex is (-10, 3, -3).
Equation 2: (x, y, z) - F = (4, -3, 1)
Expanding the equation, we get:
x - 6 = 4
y - (-2) = -3
z - 4 = 1
Simplifying the equations, we get:
x - 6 = 4 => x = 10
y + 2 = -3 => y = -5
z - 4 = 1 => z = 5
So, another possible location of the fourth vertex is (10, -5, 5).
Therefore, the two possible locations of the fourth vertex of the parallelogram are (-10, 3, -3) and (10, -5, 5).