Determine the caresian equaiton of a plane that pass through point P(1,2,2) and is perpendicular to the line (x,y,z)= 3,-1,-4 + t(-2,3,1)
If the plane is perpendicular to
L: (3,-1,-4)+t(-2,3,1)
in which the vector (-2,3,1) is parallel to the line, and therefore perpendicular to the plane P, then
the required plane is:
-2x+3y+z+k=0
where k is a constant to be determined.
Since the required plan passes through P(1,2,2), we substitute P in the plane to get:
-2(1)+3(2)+(2)+k=0
or k=-6
Therefore the plane is given by:
-2x+3y+z-6=0
Check by joining any two distinct points on the plane to form a vector and prove that the vector is perpendicular to line L.
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To determine the Cartesian equation of a plane that passes through point P(1, 2, 2) and is perpendicular to a given line, we can use the following method:
1. Find the direction vector of the given line.
The direction vector of the line is (-2, 3, 1), which is the coefficient of t in the equation of the line.
2. Find the normal vector of the plane.
Since the plane is perpendicular to the line, the normal vector of the plane will be perpendicular to the direction vector of the line. We can take the cross product of the direction vector with any other vector to find the normal vector.
Let's take the cross product of the direction vector (-2, 3, 1) with a vector perpendicular to it, such as (1, 0, 0), to get the normal vector.
Normal vector = (-2, 3, 1) x (1, 0, 0)
= [(-3 * 0) - (1 * 0), (-2 * 0) - (1 * 0), (-2 * 0) - (-2 * 1)]
= (0, 0, 2)
3. Use the normal vector and the coordinates of point P to form the Cartesian equation of the plane.
The Cartesian equation of a plane is given by the equation: Ax + By + Cz + D = 0, where (A, B, C) is the normal vector of the plane, and (x, y, z) are the coordinates of any point on the plane.
Substituting the values, we have:
0x + 0y + 2z + D = 0
To find the value of D, substitute the coordinates of point P (1, 2, 2) into the equation:
0(1) + 0(2) + 2(2) + D = 0
4 + D = 0
D = -4
Therefore, the Cartesian equation of the plane is:
2z - 4 = 0
Simplifying the equation gives:
z - 2 = 0
So, the Cartesian equation of the plane is:
z = 2