Which of the following is an equation for the plane through the point (4,-1,5) and parallel to the xy-plane.
To find the equation for a plane, we need to know a point on the plane and the normal vector to the plane. In this case, we are given that the plane is parallel to the xy-plane, which means the normal vector will have a z-component of 0.
Using this information, let's find the normal vector first. Since the plane is parallel to the xy-plane, the normal vector should be perpendicular to the xy-plane, which means it will have a direction in the z-axis.
The general form of a plane equation is Ax + By + Cz + D = 0, where (A, B, C) is the normal vector to the plane.
Since the plane is parallel to the xy-plane, its normal vector will be (0, 0, 1), as it points in the positive z-direction.
Now that we have the normal vector, we can find the equation of the plane. Plugging in the values we have:
0x + 0y + 1z + D = 0
Simplifying the equation, we get:
z + D = 0
To find the value of D, we can substitute the coordinates of the given point (4, -1, 5) into the equation:
5 + D = 0
Solving for D, we get:
D = -5
Now we can substitute the value of D back into the equation:
z + (-5) = 0
Simplifying, we get:
z - 5 = 0
Therefore, the equation for the plane through the point (4, -1, 5) and parallel to the xy-plane is:
z - 5 = 0
The xy-plane is a plane that is parallel to the z-axis. This means that all points on the plane have a z-coordinate of 0.
To find the equation of a plane parallel to the xy-plane, we can set the z-coordinate of the point (4,-1,5) to 0. This gives us the equation:
(x, y, 0) = (4, -1, 5)
Simplifying, we get:
x = 4
y = -1
Therefore, the equation for the plane through the point (4,-1,5) and parallel to the xy-plane is x = 4 and y = -1.
To find the equation of a plane parallel to the xy-plane, we can use the standard form of the equation of a plane:
Ax + By + Cz + D = 0
Since the plane is parallel to the xy-plane, the z-component of the normal vector will be zero. The normal vector of the plane is the coefficients (A, B, C) in the equation.
Given the point (4, -1, 5) and that the plane is parallel to the xy-plane, we know that the z-component of the normal vector is 0. So, the equation for the plane will be of the form:
Ax + By + D = 0
To find the values of A, B, and D, we can substitute the x and y coordinates of the given point into the equation:
A(4) + B(-1) + D = 0
Simplifying this equation, we get:
4A - B + D = 0
Since we have 3 variables (A, B, and D), we need an additional equation to solve for them. However, the problem does not provide any additional information, so we cannot find a unique solution for the equation of the plane.