What are the unit vectors that make a certain angle with two other vectors?

Find all unit vectors in ${R}^{3}$ that make an angle of $\frac{\pi}{3}$ with the vectors $\left(1 , 0 , - 1\right)$ and $\left(0 , 1 , 1\right)$.

Jul 25, 2018

$\left(\frac{1}{\sqrt{2}} , \frac{1}{\sqrt{2}} , 0\right)$.

Explanation:

Suppose that, $\vec{v} = \left(l , m , n\right)$ is the reqd. vector.

Since $| | \vec{v} | | = 1 , \therefore , {l}^{2} + {m}^{2} + {n}^{2} = 1. \ldots \ldots \ldots \ldots \ldots \ldots \left({\star}_{1}\right)$.

If $\vec{u} = \left(1 , 0 , - 1\right) \mathmr{and} \vec{w} = \left(0 , 1 , 1\right)$, then, by what is given,

$\angle \left(\vec{v} , \vec{u}\right) = \frac{\pi}{3}$.

$\therefore \vec{v} \cdot \vec{u} = | | \vec{v} | | \cdot | | \vec{u} | | \cdot \cos \left(\frac{\pi}{3}\right)$.

$\therefore \left(l , m , n\right) \cdot \left(1 , 0 , - 1\right) = 1. \sqrt{1 + 0 + 1} \cdot \frac{1}{2}$.

$\therefore l - n = \frac{1}{\sqrt{2.}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left({\star}_{2}\right)$.

Similarly, from the given $\angle \left(\vec{v} , \vec{w}\right) = \frac{\pi}{3}$, we get,

$m + n = \frac{1}{\sqrt{2.}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left({\star}_{3}\right)$.

Utilising $\left({\star}_{2}\right) \mathmr{and} \left({\star}_{3}\right) \text{ in } \left({\star}_{1}\right)$, we get,

${\left(n + \frac{1}{\sqrt{2}}\right)}^{2} + {\left(\frac{1}{\sqrt{2}} - n\right)}^{2} + {n}^{2} = 1$.

$\therefore 3 {n}^{2} = 0 , \mathmr{and} , n = 0$.

$\therefore \vec{v} = \left(l , m , n\right) = \left(\frac{1}{\sqrt{2}} , \frac{1}{\sqrt{2}} , 0\right)$, is the desired vector!