The notions about the vectors of the plane are also true in space. Two collinear vectors share the same direction (there is a number k such as one vector is k times the other), and two vectors are orthogonal if their scalar product is zero.Equation of a line (d) in spaceTo determine the equation of a line in space that has direction vector and that passes through , note that this line (d) is the set of points such that and are collinear. and are collinear if there is a number k such that , so then
The last system is called a parametric equation of the line (d).Equation of a plane P in spaceis a normal vector to a plane P, a point in this plane, and A the orthogonal projection of M on P. The vectors and are orthogonal.Thus if a point belongs to the plane P, there is a number d such as . The last equality is the equation of a plane. When you have this kind of equation for a plane (cartesian equation), you can give the coordinates of a normal vector to the plane. Read the coefficients before x, y and z. See also : geometry on fmaths.com |