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## Homework Statement

The problem states to find a unit vector that is orthogonal to [itex]\left\langle1, 1, -2\right\rangle[/itex], forms an angle of [itex]\frac{\pi}{4}[/itex] with [itex]\left\langle1, 1, 1\right\rangle[/itex] and has v

_{1}> 0.

## Homework Equations

[itex]cos\theta = \frac{\vec{u}\bullet\vec{v}}{|\vec{u}||\vec{v}|}[/itex]

## The Attempt at a Solution

Since [itex]\vec{v}[/itex] is a unit vector, I know the length is one. This means that in the above equation, the denominator will simply be [itex]|\vec{u}|[/itex].

I know that [itex]\left\langle1, 1, -2\right\rangle \bullet[/itex] [itex]\left\langle v_{1}, v_{2}, v_{3}\right\rangle = 0[/itex] so [itex]v_{3} = \frac{v_{1}+v_{2}}{2}[/itex]

[itex]cos(\frac{\pi}{4}) = \frac{\left\langle1, 1, 1\right\rangle \bullet \left\langle v_{1}, v_{2}, v_{3}\right\rangle}{\sqrt{3}} [/itex] so

[itex]v_{3} = \sqrt{\frac{3}{2}} - v_{1} - v_{2}[/itex]

I set the two equations for [itex]v_{3}[/itex] equal to each other and end up with [itex]v_{1} + v_{2} = \frac{\sqrt{6}}{3}[/itex] but I'm not sure where to go from here.