If \(\vec{a}\) = (2\(\hat{i}\) + 4 \(\hat{j}\)-6\(\hat{k}\))  and \(\vec{b}\)=(\(\hat{i}\) + 5 \(\hat{j}\)  + 7\(\hat{k}\) ) then find the unit vector in direction of vector (\(\vec{a}\) + \(\vec{b}\))

( 3\(\hat{i}\) + 9 \(\hat{j}\)  + \(\hat{k}\) ) /√91 

We have  vectors  \(\vec{a}\) = (2\(\hat{i}\) + 4 \(\hat{j}\)-6\(\hat{k}\))  and \(\vec{b}\)=(\(\hat{i}\) + 5 \(\hat{j}\)  + 7\(\hat{k}\) )   

 Then   (\(\vec{a}\) +\(\vec{b}\))    = {(2\(\hat{i}\) + 4 \(\hat{j}\)-6\(\hat{k}\)) + (\(\hat{i}\) + 5 \(\hat{j}\)  + 7\(\hat{k}\) )}

                            = ( 3\(\hat{i}\) + 9 \(\hat{j}\)  + \(\hat{k}\) )

   magnitude of (\(\vec{a}\) +\(\vec{b}\)) =√(3)² +(9)² +(1)² = √91

   The unit vector in direction of   (\(\vec{a}\) +\(\vec{b}\))=  (\(\vec{a}\) +\(\vec{b}\))/ |(\(\vec{a}\) +\(\vec{b}\))|

   So,  The unit vector in direction of  (\(\vec{a}\) +\(\vec{b}\)) =( 3\(\hat{i}\) + 9 \(\hat{j}\)  + \(\hat{k}\) ) /√91