Let be a non–zero vector in three–dimensional space whose direction cosines are denoted by

MATHEMATICS · Subjective · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Let v\mathbf{v} be a non–zero vector in three–dimensional space whose direction cosines are denoted by cos⁡α,  cos⁡β,  cos⁡γ\cos\alpha,\;\cos\beta,\;\cos\gamma. (i) State the relation connecting cos⁡α,cos⁡β\cos\alpha,\cos\beta and cos⁡γ\cos\gamma. (ii) Using the concept of a unit vector, explain why this relation must hold for every vector, irrespective of its magnitude. (iii) If a student claims that the numbers 35,  45,  65\tfrac{3}{5},\;\tfrac{4}{5},\;\tfrac{6}{5} can simultaneously serve as the direction cosines of a vector, examine the claim and justify your conclusion.

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