The question reads:
"Find the angle , where , made by the position vector with the positive directions of the axes in the following cases: (a) , (b) , (c) , (d) , (e) ."
I would have thought that the answers would be:
(a) .
This agrees with the answer sheet.
(b) I'm not sure of how best to visualise the required answer for but I suppose that it equals . And , so .
I think this must be wrong, as the answer sheet gives and .
(c) I would have thought that and . However, I'd really like to make , to communicate that the angle is measured anticlockwise from the respective positive axes. Though the desired range of angles does not allow for this, nor does it allow for anything greater than , so would be disallowed too.
The answer sheet seems to confirm these former answers.
(d) For visualising say, should one imagine hovering orthogonally above the and axes, in the region of the positive axis and visualising the angle between the vector and the axis from this "2 dimensional" viewpoint? In this case, I think that .
The answer sheet gives .
(e) I'm really not sure about this one (not that I've been particularly sure about any of the previous!), but I'd suggest that and .
The answer sheet gives and .
Could somebody help me to visualise the required angles? Also, why is it not that negative angles could be given (akin to the system used to describe the arguments of complex numbers), or angles greater than ? Is the system above amiguous, i.e. that the three angles do not imply a single specific half-line in space?
Thanks in advance for your help.
Sorry; I'm afraid that I have been victim of the tl;dr scorn in the past, so I am now a bit too careful!
I do remember the dot and cross products, but I'm working in a chapter of a textbook that precedes their discussion, so I was under the impression that it could be worked out via other methods.
But I suppose that one could use the formula you mentioned to find the angle between the given vector and the basis vectors, to give a set of solutions?