Final answer:
To find the standard Cartesian equation of a plane perpendicular to a given vector and containing a certain point, use the normal vector as the plane's coefficients and include the point to derive the equation A(x - x0) + B(y - y0) + C(z - z0) = 0.
Step-by-step explanation:
To find a standard Cartesian equation of a plane that is perpendicular to a given vector and passes through a specified point, we can use the normal vector of the plane (which is the given vector itself) and the coordinates of the point. Let's say the normal vector is = and the point is = (x0, y0, z0). The standard equation of the plane is A(x - x0) + B(y - y0) + C(z - z0) = 0. These values, A, B, and C, are the scalar components of the normal vector, corresponding to each of the x, y, and z coordinates respectively. The point (x0, y0, z0) is used to ensure that the plane passes through that point.
To construct this equation, we first identify that the normal vector is the same as the given vector that the plane is perpendicular to. In a Cartesian coordinate system, the x, y, and z components of a vector are used to describe its orientation in 3D space. Since the plane is perpendicular to the vector, the normal vector directly provides us with the coefficients A, B, and C in the plane equation.
The equation A(x - x0) + B(y - y0) + C(z - z0) = 0 represents a plane in 3-dimensional space where (A, B, C) is the direction of the normal to the plane, and the (x0, y0, z0) is a known point through which the plane passes.