Video Transcript
True or False: If 𝜃 𝑥, 𝜃 𝑦, and 𝜃 𝑧 are defined as the direction angles of vector 𝐀, then cos 𝜃 𝑥, cos 𝜃 𝑦, cos 𝜃 𝑧 are defined as the direction cosines of vector 𝐀.
We’re given that 𝜃 𝑥, 𝜃 𝑦, and 𝜃 𝑧 are the direction angles of a vector 𝐀. But what does this mean? Well, in general, given a vector 𝐀 with components 𝐴 𝑥, 𝐴 𝑦, and 𝐴 𝑧, the direction angles of 𝐀 are the angles 𝜃 𝑥, 𝜃 𝑦, and 𝜃 𝑧 that the vector makes with the 𝑥-, 𝑦-, and 𝑧-axes, respectively. And these can be written as components 𝜃 𝑥, 𝜃 𝑦, and 𝜃 𝑧.
Now, we know that in a right angle trigonometry, the cos of an angle 𝜃 is equal to the length of the side adjacent to the angle divided by the length of the hypotenuse. In our case, the length of the side adjacent to our angle 𝜃 is the 𝑥-, 𝑦-, or 𝑧-component of our vector 𝐀, and the hypotenuse is the magnitude or norm of our vector 𝐀. And hence, the cos of our direction angle 𝜃 𝑥 is the 𝑥-component of our vector 𝐀 divided by the magnitude of 𝐀, and similarly, for the cos of direction angles 𝜃 𝑦 and 𝜃 𝑧. By definition then, the direction cosines of the vector 𝐀 are the cosines of the three direction angles 𝜃 𝑥, 𝜃 𝑦, and 𝜃 𝑧. So, cos 𝜃 𝑥, cos 𝜃 𝑦, and cos 𝜃 𝑧 are defined as the direction cosines of the vector 𝐀.
Therefore, the statement “If 𝜃 𝑥, 𝜃 𝑦, 𝜃 𝑧 are defined as the direction angles of vector 𝐀, then cos 𝜃 𝑥, cos 𝜃 𝑦, cos 𝜃 𝑧 are defined as the direction cosines of vector 𝐀” is true.
