Find all the unit vectors x R3 that make an angle of /3 with each of the vectors (1

Step 1 of  2

Let \(x=\left(x_1,x_2,x_3\right)\in\mathbb R^3\) makes an angle \(\frac{\pi}{3}\) with  and

Also \(x\) is a unit vector in \(\mathbb{R}^{3}\)

Now by dot product rule

\(\left(x_{1}, x_{2}, x_{3}\right) \cdot(1,0,-1)=\|x\| \cdot\|(1,0,-1)\| \cos \frac{\pi}{3}\) ..... (1)

And \(\left(x_{1}, x_{2}, x_{3}\right) \cdot(0,1,1)=\left\|\left(x-1, x_{2}, x_{3}\right)\right\| \cdot\|(0,1,1)\| \cos \frac{\pi}{3}\) ..... (2)

Now from (1)

\(x_{1}-x_{3}=1 \cdot \sqrt{2} \cdot \frac{1}{2}\) ....(3)   [since \(\left(x=x_{1}, x_{2}, x_{3}\right)\) is unit vector]

\(x_{2}+x_{3}=1 \cdot \sqrt{2} \cdot \frac{1}{2}\) ....(3)   [since \(\left(x=x_{1}, x_{2}, x_{3}\right)\) is unit vector]

\(x_{1}-x_{3}=\frac{1}{\sqrt{2}}\)

\(x-2+x_{3}=\frac{1}{\sqrt{2}}\)

Thus  the general solution of the equation as,

\(\begin{array}{l} x_{1}=x_{3}+\frac{1}{\sqrt{2}} \\ x_{2}=-x_{3}+\frac{1}{\sqrt{2}} \\ x_{3}=x_{3}+0 \end{array}\)