The vector component of $\vec{P}=2{\hat{a}}_x-{\hat{a}}_z$ along $\vec{Q}=2{\hat{a}}_x-{\hat{a}}_y+2{\hat{a}}_z$ is:

The vector component of $\vec{P}$ along $\vec{Q}$ is given by:
$${\text{Proj}}_{\vec{Q}}\vec{P}=\left(\frac{\vec{P}\cdot \vec{Q}}{|\vec{Q}{|}^2}\right)\vec{Q}$$
First, calculate the dot product $\vec{P}\cdot \vec{Q}$:
$$\vec{P}\cdot \vec{Q}=(2)(2)+(0)(-1)+(-1)(2)=4+0-2=2$$
Next, calculate the magnitude squared $|\vec{Q}{|}^2$:
$$|\vec{Q}{|}^2=(2{)}^2+(-1{)}^2+(2{)}^2=4+1+4=9$$
Now, substitute into the projection formula:
$${\text{Proj}}_{\vec{Q}}\vec{P}=\frac{2}{9}(2{\hat{a}}_x-{\hat{a}}_y+2{\hat{a}}_z)$$