LESSON 1: Cos Angles


Cos angles can provide an efficient and helpful way to analyze vectors. For a three dimensional vector, the cos angles are the angles between the vector and the x-axis, the vector and the y-axis, and the vector and the z-axis. Each of the three cos angles are represented by a Greek letter. The angle between the vector and the x-axis is represented by the variable alpha, the angle between the vector and the y-axis is represented by the variable beta, and the angle between the vector and the z-axis is represented by gamma. These angles are significant because the inverse cos of alpha is the x-component of the vector’s unit vector,
the inverse cos of beta is the y-component of the vector’s unit vector, and the inverse cos of gamma is the z-component of the vector’s unit vector. The vice versa of this is also true; the cos of the x-component of any unit vector is the angle that vector makes with the x-axis, the cos of the y-component of any unit vector is the angle that vector makes with the y-axis, and the cos of the z-component of any unit vector is

the angle that vector makes with the z-axis.

Hopefully the graph to the left can help visualize the idea of cos angles. Let’s pretend that the vector in the graph has an x-component of 3, a y-component of 2, and a z-component of 4: v= 3 i + 2 j + 4 k. To find the cos angles of this vector begin by finding its unit vector. After finding the unit vector we can then take the inverse cos of each component to find the three cos angles for the vector. Divide each component of the vector by its magnitude to find the unit vector : .557 i + .371 j + .743 k . Now we can find the cos angles by
taking the inverse cos of each component of the unit vector. To find alpha, the angle between the vector and the x-axis, take the inverse cos of .557 to get 56.15 degrees. To find beta, the angle between the vector and the y-axis, take the inverse cos of .371 to get 68.22 degrees. To find gamma, the angle between the vector and the z-axis, take the inverse cos of .743 to get 42.01 degrees.