Find the value of £ such …

{tex}\vec a = 2\hat i + 4\hat j - 5\hat k{/tex}

{tex}\vec b = \lambda \hat i + 2\hat j + 3\hat k{/tex}

{tex}\vec a + \vec b = \left( {2 + \lambda } \right)\hat i + 6\hat j - 2\hat k{/tex}

Unit vector along

{tex}\vec a + \vec b = \frac{{\vec a + \vec b}}{{\left| {\vec a + \vec b} \right|}}{/tex}

{tex}= \frac{{\left( {2 + \lambda } \right)\hat i + 6\hat j - 2\hat k}}{{\sqrt {{{\left( {2 + \lambda } \right)}^2} + {{\left( 6 \right)}^2} + {{\left( { - 2} \right)}^2}} }}{/tex}

{tex}= \frac{{\left( {2 + \lambda } \right)\hat i + 6\hat j - 2\hat k}}{{\sqrt {{{\left( {2 + \lambda } \right)}^2} + 40} }}{/tex}

ATQ , {tex}\vec c.\left( {\vec a + \vec b} \right) = 1{/tex}

{tex}\left( {\hat i + \hat j + \hat k} \right).\left( {\frac{{\left( {2 + \lambda } \right)\hat i + 6\hat j - 2\hat k}}{{{{\left( {2 + \lambda } \right)}^2} + 40}}} \right) = 1{/tex}

{tex}\frac{{\left( {2 + \lambda } \right) + 6 - 2}}{{\sqrt {{{\left( {2 + \lambda } \right)}^2} + 40} }} = 1{/tex}

{tex}2 + \lambda + 4 = \sqrt {{{\left( {2 + \lambda } \right)}^2} + 40} {/tex}

Sq both sides,

{tex}{\lambda ^2} + 36 + 12\lambda = {\left( {2 + \lambda } \right)^2} + 40{/tex}

{tex}\lambda = 1{/tex}