Given that:
\(\vec{a} =5\hat{i} -2\hat j+5\hat k \ \\\ and \ \vec b=3\hat i+\hat k\)
\(let\ \vec a = \vec P + \vec Q\\\\
here \ \vec P \parallel \vec b \ and\ \vec Q \perp \vec b\)
\(as \ \vec P \parallel \vec b \ then \\\\
let \ \vec P =m\vec b\\\\
\vec P =3m\hat i +m\hat k...............(1)\)
\(Thus \ \vec Q =\vec a-\vec P\\\\
=5\hat i -2\hat j+5\hat k-3m\hat i -m\hat k\\\\
=(5-3m)\hat i-2\hat j+(5-m)\hat k\)
\(\vec Q \perp \vec b\ so \ \vec Q.\vec b=0\)
\([ (5-3m)\hat i-2\hat j+(5-m)\hat k ].[3\hat i+\hat k]=0\\\\
3(5-3m)-2*0+5-m=0\\\\
15-9m+5-m=0\\
10m=20\\
m=2\)
Thus :
\(\bf\vec P =6\hat i +2\hat k\\
and \ \vec Q =-\hat i-2\hat j+3\hat k\)