For the three-dimensional vector r = 1.35î + 2.56ĵ + 3.14k, determine the following.

Question

For the three-dimensional vector  r = 1.35î + 2.56ĵ + 3.14,

 determine the following.

(a) magnitude of the vector r

 

(b) angle 𝛼 between the vector r

 and the positive x axis 

(c) angle 𝛽 between the vector r

 and the positive y axis 

(d) angle 𝛾 between the vector r

 and the positive z axis 

Answer 1

a) The Pythagorean theorem looks similar in 3d: r² = x² + y² + z²

b/c/d) It would probably help you to sketch these out. In each of these situations, you will be able to draw a right-triangle where r is the hypotenuse (which you just found), and the "adjacent" side (from your trig formulas) is the value of the vector along the axis (x-axis for i, etc.).

1. cos = adj / hyp
2. cos𝛼 = 1.35 / r is one example, using the r value that you found in part a. The others will follow this pattern.