determine an equation of the plane that is perpendicular to the plane x+2y+4=0, contains the origin, and has a normal that makes an angel of 30 degrees with the z-axis.
To determine the equation of a plane that meets these criteria, we need to find the normal vector of the plane, which is perpendicular to the plane, and then use the normal vector and the coordinates of the origin to construct the equation of the plane.
1. Find the normal vector of the given plane:
Given plane equation: x + 2y + 4 = 0
The coefficients of x, y, and z in the plane equation represent the normal vector of the plane.
So, the normal vector is [1, 2, 0].
2. Convert the given angle of 30 degrees with the z-axis to a direction vector:
Given angle: 30 degrees
The z-axis points directly upwards, so we need a vector pointing upwards at a 30-degree angle to the z-axis.
Using trigonometry, we can find the direction vector:
cos(30 degrees) = adjacent/hypotenuse
cos(30 degrees) = z component of direction vector/1 (length of direction vector)
z component of direction vector = cos(30 degrees)
z component of direction vector = √3/2
3. Find the remaining components of the direction vector:
Since the plane is perpendicular to the x and y axes (given by the normal vector), the x and y components of the direction vector are 0.
So, the direction vector is [0, 0, √3/2].
4. Find the cross product of the normal vector and the direction vector:
To get a vector perpendicular to both the normal vector and the direction vector, we can take their cross product.
Normal vector = [1, 2, 0]
Direction vector = [0, 0, √3/2]
Calculating the cross product:
[1, 2, 0] × [0, 0, √3/2] = [√3/2, -√3/2, 0]
5. Determine the equation of the plane using the normal vector and the origin:
We have the normal vector for the plane, which is [√3/2, -√3/2, 0]. Since the plane passes through the origin (0, 0, 0), we can use the point-normal form of the plane equation.
The equation of the plane is:
√3/2(x - 0) - √3/2(y - 0) + 0(z - 0) = 0
Simplifying:
√3/2(x) - √3/2(y) = 0
Multiplying through by 2 to eliminate the fraction:
√3(x) - √3(y) = 0
This is the equation of the plane that is perpendicular to the plane x + 2y + 4 = 0, contains the origin, and has a normal that makes an angle of 30 degrees with the z-axis.
To determine an equation of the plane that is perpendicular to the plane x+2y+4=0, contains the origin, and has a normal that makes an angle of 30 degrees with the z-axis, we need to follow these steps:
1. Find the normal vector of the given plane, x + 2y + 4 = 0.
The coefficients of x, y, and z in the equation represent the components of the normal vector.
So, the normal vector of the given plane is (1, 2, 0).
2. Find a vector that is perpendicular to the normal vector and the z-axis.
Since the normal vector makes an angle of 30 degrees with the z-axis, we can use trigonometry to find the components of this vector.
The z-component of the vector will be the projection of the normal vector onto the z-axis, which is given by n * cos(theta), where n is the magnitude of the normal vector (sqrt(1^2 + 2^2 + 0^2)) and theta is the angle between the normal vector and the z-axis (30 degrees).
Therefore, the z-component = sqrt(5) * cos(30 degrees) = sqrt(5) * sqrt(3) / 2 = sqrt(15) / 2.
The x and y components of the vector will both be zero since it is perpendicular to the z-axis.
So, the vector that is perpendicular to the normal vector and the z-axis is (0, 0, sqrt(15) / 2).
3. Use the normal vector and the point (0, 0, 0) (which is the origin) to write the equation of the plane.
The equation of a plane is given by Ax + By + Cz + D = 0, where (A, B, C) is the normal vector and (x, y, z) is any point on the plane.
Plugging in the values, we get:
(1 * x) + (2 * y) + (0 * z) + D = 0.
Since the origin (0, 0, 0) lies on the plane, we can substitute the values into the equation. Therefore, we have:
(1 * 0) + (2 * 0) + (0 * 0) + D = 0.
This simplifies to D = 0.
So, the equation of the plane that is perpendicular to x + 2y + 4 = 0, contains the origin, and has a normal that makes an angle of 30 degrees with the z-axis is:
x + 2y = 0.