Let b → = i ^ + j ^ + λ k ^ , λ ∈ ℝ . If a → is a vector such that a…

Let $\overrightarrow{\mathrm{a}}=x\hat{i}+y\hat{j}+z\hat{k}$

So,$\left|\begin{array}{lll}\hat{i}& \hat{j}& \hat{k}\\ x& y& z\\ 1& 1& \lambda \end{array}\right|=\hat{i}\left(\lambda y-z\right)+\hat{j}\left(z-\lambda x\right)+\hat{k}\left(x-y\right)$

On comparing with $13\hat{i}-\hat{j}-4\hat{k}$ we get,

$\Rightarrow \lambda y-z=13,z-\lambda x=-1,x-y=-4$

and $x+y+\lambda z=-21$

$\Rightarrow$ Clearly, $\lambda =3,x=-2,y=2$ and $z=-7$

So, $\overrightarrow{\mathrm{b}}-\overrightarrow{a}=3\hat{i}-\hat{j}+10\hat{k}$ and $\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{a}}=-\hat{i}+3\hat{j}-4\hat{k}$

$\Rightarrow \left(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}\right)·\left(\hat{k}-\hat{j}\right)+\left(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{a}}\right)·\left(\hat{i}-\hat{k}\right)=11+3$

$=14$