Question

determine the components of the unit vector u=(a,b,c) which has directional angles alpha = 100degrees, and beta=64.5degrees?

Answer 1

We can position a coordinate system at one end of a vector (a,b,c) as seen in the diagram. Note this is not assumed to be a unit vector.

The x-coordinate is then\[a=h \cos \alpha\]The y-coordinate,\[b=h \sin \alpha\]The z-coordinate,\[c=h \tan \beta\]So the (non-unit vector), (a,b,c) has the form,\[(h \cos \alpha, h \sin \alpha, h \tan \beta)\]We form the unit vector by dividing it through by the magnitude, the magnitude being,\[\sqrt{a^2+b^2+c^2}=\sqrt{h^2 \cos^2 \alpha + h^2 \sin^2 \alpha + h^2 \tan^2 \beta}\]\[=h \sqrt{1+\tan^2 \beta}=h \sec \beta\]So the unit vector is\[\frac{1}{h \sec \beta}h(\cos \alpha, \sin \alpha, \tan \beta)=(\cos \alpha \cos \beta, \sin \alpha \cos \beta, \sin \beta)\]Substituting in alpha = 100 degrees and beta = 64.5 degrees should give you,\[(a,b,c)=(-0.0748,0.424, 0.903) \]to three significant figures.