Linear Algebra - Use angles between vectors to find other vectors

[Heute]

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 = (x_{1},x_{2},x_{3})

We need X*Y = cos(\pi/4) |x| |y|
and X*Z = cos(\pi/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= x_{1} + x_{3} = srt(2)/2 * sqrt(2) = 1
X*Z = x_{2} = 1/2 * sqrt(2) = sqrt(2)/2

Solving for x_{1} and x_{2} in terms of x_{3} we get:

X = (1, 1/2, 0) + x_{3}(-1, 0, 1)

Problem: X is not a unit vector for all x_{3}, 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 x_{1}^2 + x_{2}^2 + x_{3}^2 = 1, but I'm not sure how to handle such a nonlinear constraint!

 

[vela]

The Attempt at a Solution

let X = (x_{1},x_{2},x_{3})

We need X*Y = cos(\pi/4) |x| |y|
and X*Z = cos(\pi/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= x_{1} + x_{3} = srt(2)/2 * sqrt(2) = 1
X*Z = x_{2} = 1/2 * sqrt(2) = sqrt(2)/2

Why is there a factor of $\sqrt{2}$ in the equation for $\vec{x}\cdot\vec{z}$?

 

[Heute]

Apologies! That sqrt(2) should be a 1 (the length of Z). I copied that sqrt(2) from the memory of another similar problem I was working on. I'll edit the original post!

This doesn't, however fundamentally change things. The question still remains.

 

[Heute]

Apparently, I can't edit the original post, so I'll re-write it here:

The Attempt at a Solution

let X = (x_{1},x_{2},x_{3})

We need X*Y = cos(\pi/4) |x| |y|
and X*Z = cos(\pi/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= x_{1} + x_{3} = srt(2)/2 * sqrt(2) = 1
X*Z = x_{2} = 1/2 * 1 = 1/2

Solving for x_{1} and x_{2} in terms of x_{3} we get:

X = (1, 1/2, 0) + x_{3}(-1, 0, 1)

Problem: X is not a unit vector for all x_{3}, 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 x_{1}^2 + x_{2}^2 + x_{3}^2 = 1, but I'm not sure how to handle such a nonlinear constraint!

 

[vela]

You just need to find the value or values of x₃ for which X has unit length. Solve the equation X²=1 for x₃.