- #1
Illania
- 26
- 0
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 v1 > 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.