If , , are the lengths of the sides of opposite to the angles , and respectively, prove using vector methods that
We have, by the triangle law,
Adding , and we have
on both sides, we have
Find three-dimensional vectors and satisfying the relations
A reference frame for the vectors has not been specified; therefore, it is up to us to choose a reference frame and then use it consistently and evaluate the required vectors in that reference frame.
Assume to be along the -direction, i.e.
Now we step by step use all the given relations to determine the unknown constraints:
Notice that , and are three equations in four unknowns. To get over this problem (it is not a problem actually! There will be an infinite set of vectors satisfying the given constraints. We have to find any one of them), when we chose to be along the -axis, we could also have adjusted the co-ordinate frame, so that and lie in the plane. This can always be done; since it is upto us to choose the frame of reference, we chose it so that the plane co-insides with the plane of and .
How does this help? Now we’ll have one unknown less, since the -component of is zero, i.e., .
Thus, , and reduce to
Thus, the three dimensional vectors that satisfy the given constraints can be
To emphasize once again, we were required to find vectors satisfying the given constraints. This meant that absolute positions of the vectors were not important; what mattered was their relative sizes and orientation; and thus the coordinate axes was our choice. We selected it in a way which made the calculations most convenient.