To find the equation of a plane through three points, we can use the formula:
Ax + By + Cz = D
where A, B, C, and D are constants to be determined.
Let's start by finding the direction vectors of two lines on the plane. We can do this by subtracting the coordinates of one point from the other two points.
Direction vector of the first line:
(0, -1, -1) - (0, 1, 0) = (0, -2, -1)
Direction vector of the second line:
(1, 2, 1) - (0, 1, 0) = (1, 1, 1)
Now, we can find a normal vector to the plane by taking the cross product of the two direction vectors.
Normal vector to the plane:
(0, -2, -1) x (1, 1, 1) = (1, -1, 2)
Now, let's substitute one of the given points (0, 1, 0) into the equation Ax + By + Cz = D to find D.
1*0 + (-1)*1 + 2*0 = D
-1 = D
Therefore, the equation of the plane is:
x - y + 2z = -1
Find equation of a plane through points
(0, 1, 0), (0, -1, -1) and (1, 2, 1)
1 answer