Perpendiculars to a line, distances from a line and the equation of a plane

A plane is fixed by one point A of it and one direction perpendicular to it. A normal vector n is any non-zero vector perpendicular to the plane (in Arabic المتجه العمودي على المستوى). A point P with position vector r lies in the plane exactly when AP is perpendicular to n, that is n · (r − A) = 0, so n · r = n · A. This is the vector equation of the plane. With n = (a, b, c), A = (x₁, y₁, z₁) and r = (x, y, z), it reads a(x − x₁) + b(y − y₁) + c(z − z₁) = 0, the standard form. Multiplying out gives ax + by + cz + d = 0 with d = −(ax₁ + by₁ + cz₁), the general form. In the general form the coefficients of x, y and z are the components of a normal vector. They are not changed by multiplying the whole equation by a number. To test whether a point is in the plane, put its coordinates into the equation. Special planes: z = k has normal (0, 0, 1) and is parallel to the xy-plane; if the coefficient of y is 0, as in 5x + 2z = 10, the normal has b = 0, and the plane is parallel to the y-axis. A plane has no 'slope'; the normal vector is what tells us its tilt.