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Homework Statement
Find all the unit vectors X element of R^3 that make an angle of pi/4 radians with vector Y = (1,0,1) and an angle of pi/3 radians with vector Z = (0,1,0)
Homework Equations
For any two vectors X and Y element of R^n, the dot-prodict of X and Y is equals to the length of X times the length of Y times the cosine of the angle between X and Y. That is:
X*Y = cos(t)|x||y|
The Attempt at a Solution
let X = ([itex]x_{1}[/itex],[itex]x_{2}[/itex],[itex]x_{3}[/itex])
We need X*Y = cos([itex]\pi[/itex]/4) |x| |y|
and X*Z = cos([itex]\pi[/itex]/3)|x||z|
We want the length of X, that is |x|, to be 1 so I'll assume that it is 1 for now (this could be a bad idea).
X*Y= [itex]x_{1}[/itex] + [itex]x_{3}[/itex] = srt(2)/2 * sqrt(2) = 1
X*Z = [itex]x_{2}[/itex] = 1/2 * sqrt(2) = sqrt(2)/2
Solving for [itex]x_{1}[/itex] and [itex]x_{2}[/itex] in terms of [itex]x_{3}[/itex] we get:
X = (1, 1/2, 0) + [itex]x_{3}[/itex](-1, 0, 1)
Problem: X is not a unit vector for all [itex]x_{3}[/itex], so we haven't really found a formula for "all unit vectors" which fulfill the initial requirements. This is where I'm stuck! I thought about including another variable for |x| in the restrictions and adding a further restriction that [itex]x_{1}[/itex]^2 + [itex]x_{2}[/itex]^2 + [itex]x_{3}[/itex]^2 = 1, but I'm not sure how to handle such a nonlinear constraint!