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.