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# If vector $\overrightarrow a$ is unit vector and it makes angle $\large\frac{\pi}{4}$ with $\hat k$ and $\overrightarrow a+\hat i+\hat j$ is also a unit vector, then $\overrightarrow a=?$

$\begin{array}{1 1} (A) \large\frac{1}{2}\hat i+\large\frac{1}{2}+\hat j+\large\frac{1}{\sqrt 2}\hat k \\ (B) -\large\frac{1}{2}\hat i+\large\frac{1}{2}+\hat j+\large\frac{1}{\sqrt 2}\hat k\\ (C) -\large\frac{1}{2}\hat i-\large\frac{1}{2}+\hat j+\large\frac{1}{\sqrt 2}\hat k\\ (D) \large\frac{1}{2}\hat i-\large\frac{1}{2}+\hat j+\large\frac{1}{\sqrt 2}\hat k \end{array}$

Let $\overrightarrow a=\alpha \hat i+\beta \hat j+\gamma \hat k$
Given: $|\overrightarrow a|=1\Rightarrow\:\alpha ^2+\beta^2+\gamma^2=1.......(i)$
Given: $\overrightarrow a.\hat k=cos\large\frac{\pi}{4}=\frac{1}{\sqrt 2}$
$\Rightarrow\:\gamma=\large\frac{1}{\sqrt 2}$
$\Rightarrow\:\alpha^2+\beta^2=\large\frac{1}{2}........(ii)$
Also it is given that $\overrightarrow a+\hat i+\hat j$ is a unit vector.
$\Rightarrow\:(\alpha+1)^2+(\beta+1)^2+\large\frac{1}{2}$$=1......(iii) From (ii)\:\:and \:\:(iii) we get \alpha+\beta=-1....(iv) Solving (ii) and (iv) we get \alpha=\beta=-\large\frac{1}{2} \therefore\:\overrightarrow a=-\large\frac{1}{2}$$\hat i-\large\frac{1}{2}$$+\hat j+\large\frac{1}{\sqrt 2}$$\hat k$