Let ^ u = u _ 1 ^ i + u _ 2 ^ j + u _ 3 ^ k be a unit vector in ℝ ^...

As | \overset{^}{w} | = | \overset{^}{u} | = 1 = | \overset{^}{u} \times \overset{\to}{v} |
Let $ϕ$ is angle between \overset{^}{w} \text{ and } ( \overset{^}{u} \times \overset{\to}{v} )
\left|\overset{^}{w}\right|\left|\overset{^}{u} \times \overset{\to}{v}\right|cos ϕ = 1 ⇒ ϕ = 0$<br />$⇒ \overset{^}{u} \times \overset{\to}{v} = \overset{^}{w}$<br />$⇒$There may be infinite vectors$\overset{\to}{v}$Satisfying this condition<br />If$\overset{^}{u}$his in$xy$plane :$\overset{^}{u} \times \overset{\to}{v} = \left|\begin{cases}\overset{^}{i} & \overset{^}{j} & \overset{^}{k} \\ {u}_{1} & {u}_{2} & 0 \\ {v}_{1} & {v}_{2} & {v}_{3}\end{cases}\right|$= \overset{^}{w}
$⇒ {u}_{3} = 0$
Let \overset{\to}{v} = {v}_{1} \overset{^}{i} + {v}_{2} \overset{^}{j} + {v}_{3} \overset{^}{k}
\overset{^}{w} = \left({u}_{2}{v}_{3}\right)\overset{^}{i} - \left({u}_{1}{v}_{3}\right)\overset{^}{j} + \left({u}_{1}{v}_{2} - {u}_{2}{v}_{1}\right)\overset{^}{k} = \dfrac{1}{\sqrt{6}} \overset{^}{i} + \dfrac{1}{\sqrt{6}} \overset{^}{j} + \dfrac{2}{\sqrt{6}} \overset{^}{k}
${u}_{2}{v}_{3} = \dfrac{1}{\sqrt{6}}, - u{}_{1}{v}_{3} = \dfrac{1}{\sqrt{6}} ⇒ \left|{u}_{1}\right| = \left|{u}_{2}\right|$
If \overset{^}{u} his in $xz$ plane $⇒ {u}_{2} = 0$ : \overset{^}{u} \times \overset{\to}{v} = \left|\begin{cases}\overset{^}{i} & \overset{^}{j} & \overset{^}{k} \\ {u}_{1} & 0 & {u}_{3} \\ {v}_{1} & {v}_{2} & {v}_{3}\end{cases}\right|
\overset{^}{w} = \left(- {v}_{2}{u}_{3}\right)\overset{^}{i} - \left({u}_{1}{v}_{3} - {u}_{3}{v}_{1}\right)\overset{^}{j} + \left({u}_{1}{v}_{2}\right)\overset{^}{k} = \dfrac{1}{\sqrt{6}} \overset{^}{i} + \dfrac{1}{\sqrt{6}} \overset{^}{j} + \dfrac{2}{\sqrt{6}} \overset{^}{k}
$- {v}_{2}{u}_{3} = \dfrac{1}{\sqrt{6}}, {u}_{1}{v}_{2} = \dfrac{2}{\sqrt{6}}$
$⇒ 2\left|{u}_{3}\right| = \left|{u}_{1}\right|