Please find the coordinates of this vector d=(a,b,c), that makes an angle of 30 degrees with the y-axis, 60 degrees with he z axis and is perpendicular to the x-axis. Is the answer distinct?    

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mlehuzzah eNotes educator | Certified Educator

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First, picture, for a moment, in 2D, the following four vectors:

<1,1>, <-1,1> , <-1,-1>, <1,-1>

They each make a 45 degree angle with the x-axis (and a 45 degree angle with the y-axis). Moreover, we could stretch our vector:
<2,2> , <3,3>, etc, also make a 45 degree angle with each of the x and y axes.

So if we were looking for a vector (in 2D) that made a 45 degree angle with the x and y axes, the answer would not be unique.

The nonuniqueness comes from two places:

1) To be 45 degrees from the x-axis means to make a 45 degree angle with EITHER of <1,0> or <-1,0>. That is,...

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