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` find a unit vector parallel to the vector `2 vec a- vec b+3 vec c` .

Explanation

Given that, `vec(a) = hat(i)+hat(j)+hat(k), vec(b) = 2hat(i) - hat(j) + 3hat(k)` and `vec(c ) = hat(i) - 2 hat(j) + hat(k)`
Let `vec(X) = 2a -b + 3x`
`= 2 (hat(i) + hat(j) + hat(k)) - (2hat(i) - hat(j) + 3hat(k)) + 3(hat(i) - 2hat(j) + hat(k)) = 3hat(i)-3hat(j)+ 2hat(k)`
Now, `|vec(X)| = |3 hat(i)-3hat(j) + 2hat(k)|`
`= sqrt(3^(2) + (-3)^(2) + 2^(2))= sqrt(22)`
Therefore, unit vector along the vector `vec(X)`
`hat(X)=(vec(X))/(|vecX|) = (3hat(i) - 3hat(j) + 2hat(k))/(sqrt(22))`
`=((3)/(sqrt(22))hat(i) - (3)/(sqrt(22))hat(j) + (2)/(sqrt(22))hat(k))`