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 systemís 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 planeís 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

 

†††††††††††