A vector → v in the first octant is inclined to the x axis at 60 °...

Direction cosines $l, mandn$ of given vector $\overset{\to}{v}$,

$l = \cos \left({60}^{∘}\right) = \dfrac{1}{2}$

$m = \cos \left({45}^{∘}\right) = \dfrac{1}{\sqrt{2}}$

$n = \cos \left(\gamma\right)$

\overset{\to}{v} = \cos 60 ° \overset{^}{i} + \cos 45 ° \overset{^}{j} + cos\gamma\overset{^}{k} . . . . ( 1 )

We know,

${l}^{2} + {m}^{2} + {n}^{2} = 1$

$⇒ \dfrac{1}{4} + \dfrac{1}{2} + \cos^{2}\gamma = 1 ( \gamma \to$Acute$)$

$⇒ cos\gamma = \dfrac{1}{2}$

$⇒ \gamma = 60 °$

Equation of plane is $\dfrac{1}{2}\left(x - \sqrt{2}\right) + \dfrac{1}{\sqrt{2}}\left(y + 1\right) + \dfrac{1}{2}$\left(z - 1\right) = 0$</p><p>$⇒ \dfrac{x}{2} + \dfrac{y}{\sqrt{2}} + \dfrac{z}{2} = \dfrac{1}{2}$</p><p>$⇒ x + \sqrt{2}y + z = 1$</p><p>\left(a, b, c\right)$ lies on it,

$⇒ a + \sqrt{2}b + c = 1$