Review of Analytic Geometry

Plane and Solid Geometry

Direction Cosines of a Line: To define the orientation of a line in space, we need to know the angles that it forms with the axes of a given coordinate system. In most situations, these angles are difficult to measure, so we make use of analytic geometry and trigonometry to determine them.

Let's consider two arbitrary points P1 and P2. We use the Cartesian coordinates (x,y,z) to identify the position of each point. Then we consider a straight line passing through P1 and P2. The line segment between these two points has a length of d. To calculate the distance d between points P1 and P2, we use the following relationship


Based on the trignometric relationship between line segment d and its projection along each axis, we determine the direction cosines of the line as


where α, β, γ are the angles that the line makes with the positive x, y, z axes, respectively. A unique characteristic of the direction cosines is that they satisfy the condition


Therefore, if we know any two of the three angles, we can use Eq. (10) to find the third.

figure Animation of Cartesian Components and Direction Angles of a Vector

  • 1. Move the mouse over the box for the angles to appear
  • 2. Click on the button for the angle and component you wish to view
  • 3. Move the mouse to one of the corners of the box for the vector to appear
  • 4. The 0o and 90o direction angles are identified in each case
  • 5. The non-zero Cartesian components of the vector are shown at the bottom

  • Plane and Solid Geometry