To find where the plane intersects the coordinate axes, set two of the variables to zero and solve for the third to find each axis intercept: x-intercept = (-d/a, 0, 0), y-intercept = (0, -d/b, 0), and z-intercept = (0, 0, -d/c).
To find the coordinates of the points where the plane ax + by + cz + d = 0 meets the three coordinate axes, we follow a simple procedure.
We calculate each intercept by setting the other two variables to zero and solving for the remaining one.
To find the x-intercept, set y = 0 and z = 0 in the equation. Solving ax + d = 0 for x gives us the x-intercept (x, 0, 0) where x = -d/a.
For the y-intercept, set x = 0 and z = 0. Solving by + d = 0 for y yields the y-intercept (0, y, 0) where y = -d/b.
To find the z-intercept, set x = 0 and y = 0. Solving cz + d = 0 for z provides the z-intercept (0, 0, z) where z = -d/c.
Considering a² + b² + c² ≠ 0 ensures that the plane is not parallel to any of the coordinate axes, allowing us to find distinct intercepts on each axis.