If  \(\vec{a}\ =\ \hat{i}\ +\ 2\hat{j}\ +\ 2\hat{k}\)  and  \(\vec{b}\ =\ 3\hat{i}\ +\ 5\hat{j}\ +\ \sqrt{2}\hat{k}\) then a vector in the direction of \(\vec{a}\) and having magnitude as \(|\vec b|\) is 

Concept:

Vector \(\vec{r}\) of magnitude |p| in the direction of \(\vec{q}\) is given by

\(\vec{r}\ =\ |p|.\hat{q}\ =\ |p|.\frac{\hat{q}}{|q|}\)

​If  \(\vec{a} = {a_1}\hat{i} \ + \ {a_2}\hat{j} \ + \ {a_3}\hat{k}\) then magnitude of ​\(\vec{a}\) is written as​​

\(|\vec{a}|\ =\ \sqrt{a_1^2\ +\ a_2^2\ +\ a_3^2}\)

Calculation:

Let the required vector is \(\vec{c}\)

\(\vec{a}\ =\ \hat{i}\ +\ 2\hat{j}\ +\ 2\hat{k}\)

\(\vec{b}\ =\ 3\hat{i}\ +\ 5\hat{j}\ +\ \sqrt{2}\hat{k}\)

A vector in the direction a and magnitude |b| is given by

\(\vec{c}\ =\ |b|.\hat{a}\ =\ |b|.\frac{\hat{a}}{|a|}\)

\(\vec{c}\ =\ \sqrt{3^2\ +\ 5^2\ +\ (\sqrt{2})^2}.\frac{\hat{\hat{i}\ +\ 2\hat{j}\ +\ 2\hat{k}}}{\sqrt{1^2\ +\ 2^2\ +\ 2^2}}\)

\(\vec{c}\ =\ 6.\frac{\hat{\hat{i}\ +\ 2\hat{j}\ +\ 2\hat{k}}}{3}\)

\(\vec{c}\ =\ 2({\hat{\hat{i}\ +\ 2\hat{j}\ +\ 2\hat{k}}})\)

Hence, option 4 is correct.