I presume you know: If <A, B, C> is a vector normal to a plane and [itex](x_0, y_0, z_0)[/itex] is a point in the plane, then the Cartesian equation of the plane is [itex]A(x- x_0)+ B(y- y_0)+ C(z- z_0)= 0[/itex].yy205001 said:Homework Statement
If the line "l" is given by the equations 2x-y+z=0, x+z-1=0, and if M is the point (1,3,-2), find a Cartesian equation of the plane.
a) passing through M and l
<2, 1, 1> is a vector normal to 2x- y+ z= 0. <1, 0, 1> is a vector normal to x+ z= 1. Their cross product is normal to both so normal to any plane perpendicular to the line. You have a normal vector and a point in the plane.b) passing through M and orthogonal to l
Homework Equations
(r-r0)n=0The Attempt at a Solution
I expanded the equation above, so i got rn=r0n
then, (x,y,z)n=(1,3,-2)n.
And i don't know how to get the normal vector from those two equations
The Cartesian equation of a plane is an equation that represents all the points on a plane in a three-dimensional coordinate system. It is in the form of ax + by + cz + d = 0, where a, b, and c are the coefficients of the x, y, and z variables, and d is a constant term.
The Cartesian equation of a plane can be derived using the vector equation of a plane, which is r · n = d. Here, r is the position vector of a point on the plane, n is the normal vector to the plane, and d is the distance from the origin to the plane along the direction of the normal vector.
The coefficients in the Cartesian equation of a plane represent the direction and orientation of the plane. The coefficients a, b, and c are known as the direction cosines, and they determine the direction of the normal vector to the plane. The constant term d represents the distance of the plane from the origin.
Yes, the Cartesian equation of a plane can also be written in the form of (x - x0)/a = (y - y0)/b = (z - z0)/c, where (x0, y0, z0) is a point on the plane. This form is known as the symmetric form of the equation.
The Cartesian equation of a plane is used in various fields of science and engineering, such as physics, geometry, and computer graphics. It is used to represent and analyze the position and orientation of objects in three-dimensional space. It is also used in navigation systems, computer simulations, and 3D modeling.