Final answer:
To find the equation of the plane passing through the given points, we can use the equation of a plane in 3D space. The equation of the plane passing through the points (4,1,1), (-1,-5,-10), and (-5,5,-4) is satisfied by all points in 3D space, making it a unique plane. To determine where the line crosses the z-axis, we need to find the point of intersection between the line and the z-axis.
Step-by-step explanation:
To find the equation of the plane passing through the points (4,1,1), (-1,-5,-10), and (-5,5,-4), we can use the equation of a plane in 3D space:
A(x - x1) + B(y - y1) + C(z - z1) = 0
where A, B, and C are the coefficients of the plane equation and (x1, y1, z1) is a point on the plane.
Substituting the coordinates of one of the points into the equation, we can find the values of A, B, and C. Let's use the point (4,1,1):
A(4 - 4) + B(1 - 1) + C(1 - 1) = 0
0 = 0
Since A, B, and C are all zero, the equation simplifies to 0 = 0 which is always true. This means that the equation of the plane passing through those three points is satisfied by all points in 3D space, making it a unique plane.
To determine where the line crosses the z-axis, we need to find the point of intersection between the line and the z-axis. The z-axis is given by the equation x = 0 and y = 0. Substituting these values into the equation of the line, we can find the z-coordinate of the point of intersection. Let's calculate it:
4A + B + C = 0
Since A, B, and C are all zero, the equation simplifies to 0 = 0 which is always true. This means that the equation of the plane passing through those three points is satisfied by all points in 3D space, making it a unique plane.