if \(\vec{a}\)= \(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\),  \(\vec{b}\)= 2\(\hat{i}\) -\(\hat{j}\)+ 3\(\hat{k}\) and \(\vec{c}\) = \(\hat{i}\) - 2 \(\hat{j}\) + \(\hat{k}\) then, find a unit vector parallel to the vectors 2\(\vec{a}\)  -\(\vec{b}\)+ 3\(\vec{c}\)

(3\(\hat{i}\)- 3\(\hat{j}\)+ 2\(\hat{k}\))/ √22

We have \(\vec{a}\)= \(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\),  \(\vec{b}\)= 2\(\hat{i}\) -\(\hat{j}\)+ 3\(\hat{k}\) and \(\vec{c}\) = \(\hat{i}\) - 2 \(\hat{j}\) + \(\hat{k}\) 

then  2\(\vec{a}\)  -\(\vec{b}\)+ 3\(\vec{c}\)  = 2( \(\hat{i}\)+  \(\hat{j}\)+ \(\hat{k}\)) - (2\(\hat{i}\)- \(\hat{j}\) ̂+ 3\(\hat{k}\)) + 3(\(\hat{i}\)- 2 \(\hat{j}\)+ \(\hat{k}\))

                             = (3\(\hat{i}\)- 3\(\hat{j}\)+ 2\(\hat{k}\)) 

|2\(\vec{a}\)  -\(\vec{b}\)+ 3\(\vec{c}\)| = √(3)² + (-3)² +(2)² = √22

Hence, the unit vector along 2\(\vec{a}\)  -\(\vec{b}\)+ 3\(\vec{c}\)  is

= (2\(\vec{a}\)  -\(\vec{b}\)+ 3\(\vec{c}\))/|2\(\vec{a}\)  -\(\vec{b}\)+ 3\(\vec{c}\)| = (3\(\hat{i}\)- 3\(\hat{j}\)+ 2\(\hat{k}\))/ √22