For given vectors , `a=2hat(i)-j+2hat(k) and b=-hat(i)+j-hat(k)`, find the unit vector in the direction of the vector `vec(a)+vec(b)`.

Explanation

The given vectors are `a=2hat(i)-hat(j)+2hat(k) and b=-hat(i)+j-hat(k)`.
`therefore vec(a)+vec(b)=(2hat(i)-hat(j)+2hat(k))+(-hat(i)+hat(j)-hat(k))`
Two vectors can be added by adding `hat(i), hat(j) and hat(k)` , components.
`therefore vec(a)+vec(b)=[2hat(i)+(-hat(i))]+[(-hat(j))+hat(j)]+[(2hat(k))+(-hat(k))]=(2hat(i)-hat(i))+(-hat(j)+hat(j))+(2hat(k)-hat(k))=hat(i)+0hat(j)+hat(k)=hat(i)+hat(k)`
Comparing with `X=xhat(i)+yhat(j)+zhat(k)`, we get `x=1,y=0,z=1`
`therefore`Magnitude `|vec(a)+vec(b)|=sqrt(x^2+y^2+z^2)=sqrt(1^2+0^2+1^2)=sqrt(2)`
Hence, the unit vector in the direction of `(a+b)`,
`((vec(a)+vec(b)))/(|vec(a)+vec(b)|)=((vec(i)+vec(k)))/(sqrt(2))=(1)/(sqrt(2))hat(i)+(1)/(sqrt(2))hat(k)`.