Using Direction Cosines

 

            Direction cosines are given short shrift in our schools.  One thing I learned incidentally during my PhD was to always avoid sines and cosines whenever possible and instead use direction cosines.  In planar geometry, sines and cosines are just fine; but in solid geometry, they are next to impossible to use.

            The direction cosines of a vector are merely the cosines of the angles that the vector makes with the x, y, and z axes, respectively.  We label these angles  (angle with the x axis),  (angle with the y axis), and  (angle with the z axis); and we define

 

           

           

           

 

The cosines of the angles can be found by taking dot products.  Thus, if we have a unit vector given by , the direction cosines are

 

           

           

           

 

Since the vector is a unit vector, it also follows that

 

           

 

Definition of a Line

            A vector is usually interpreted as passing through a coordinate system’s origin.  It provides direction information but not location information.  We can define a line parametrically by starting with a vector and adding an initial position .

 

           

           

           

 

Definition of a Plane

            The orientation of a plane is defined by means of the direction of a perpendicular vector .  The position of the plane is added by means of some initial conditions .  Thus, a plane is given by the equation

 

           

 

Usually, the right side of the equation is brought to the left-hand side to obtain

 

           

 

or more commonly

 

           

 

Equation of a Plane Passing Through Three Points

            Given three points in space, it is easy to compute the equation of the plane passing through them.  Let us define the three points by means of three vectors, ,, and .  Let us choose the third vector as a reference.  Then, two vectors that lie in the plane are

 

           

           

 

The normalized cross product of these two vectors yield the perpendicular to the plane:

 

           

 

Now we have the perpendicular to the plane, and we can continue with Definition of a Plane using any of the three points as an initial condition for the plane’s location.

 

Equation of a Line Perpendicular to a Plane and Passing Through a Point

            Given the equation of the plane

 

           

 

we recall that , , and , which are the direction cosines of a vector perpendicular to the plane.  (Note that we had to normalize the coefficients so that .)  Thus, if we know that the line also passes through , we can proceed to Definition of a Line to obtain the parametric representation of the line:

 

           

           

           

 

Shortest Distance from a Point to a Line

            First, we define the point in space as , the line as having direction cosines , and a point on the line as .  A vector from the point on the line to the point in space is

 

           

 

The distance between the point and line can be found with the aid of the cross product of direction cosines and .

 

           

 

           

 

This technique works because the line forms one side of a triangle,  describes the hypotenuse, and we want to know the length of the third side.  The included angle of the line and  has a sine given by the cross product.  Multiplying by the length of the hypotenuse yields the distance.

 

Shortest Distance from a Point to a Plane

            The shortest distance from a point  to a plane will be along a vector that is perpendicular to the plane.  Given the equation of the plane

 

           

 

we recall that , , and  are the direction cosines of a vector perpendicular to the plane.  (Note that we had to normalize the coefficients so that .)  We choose the point as the reference for the vector.  The line that passes through the point and is perpendicular to the plane is

 

           

           

           

 

Distance along this line is measured by

 

           

           

           

 

or

           

 

Thus, we have only to find t when the line intercepts the plane.  t can be positive or negative depending on whether the vector points toward or away from the plane.  We usually choose the sign so that distance is positive.

            The line and plane intercept when

 

           

 

Solving for t yields

 

           

 

or

 

           

 

Thus, the distance from the point to the plane is

 

           

 

Rotation of a Coordinate System

            Given one coordinate system, , a new system can be devised by rotating the old system.  One way to do this is through quaternions.  Here, we use direction cosines.  We assume that we know the direction cosines of each of the axes in the new system relative to the old system.  The direction cosines of the new x axis are .  Similar notation exists for the other axes.  The new coordinates can be described in terms of the old coordinates as

 

           

 

The old coordinates can be described in terms of the new coordinates as