Express the vector vector a = 5i - 2j + 5k as the sum of two vectors such that one is parallel to the vector vector b = 3i + k

Question

Express the vector \(\vec{a}=5\hat{i}-2\hat{j}+5\hat{k}\) as the sum of two vectors such that one is parallel to the vector \(\vec{b}=3\hat{i}+\hat{k}\) and the other is perpendicular to \(\vec{b}.\)

Answer 1

Let \(\vec{a}=\vec{c}+\vec{d}\) such that \(\vec{c}\) is parallel to \(\vec{b}\) and \(\vec{d}\) is perpendicular to \(\vec{b}\).

Again, \(\because\) \(\vec{d}\) is perpendicular to \(\vec{b}.\)

Answer 2

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 :

\(\bf\vec P =6\hat i +2\hat k\\ and \ \vec Q =-\hat i-2\hat j+3\hat k\)