Correct Answer - Option 1 : 3î - ĵ + 3k̂
Calculation:
\(\rm \vec a =\hat i +\hat j +\hat k,\; \vec b =\hat i -\hat j + \hat k\) and c = î - ĵ - k̂
Given: vector \(\rm \vec v\) in the plane of \(\rm \vec a\) and \(\rm \vec b\)
Therefore, \(\rm \vec v = \vec a + λ \vec b\)
⇒ \(\rm \vec v =(\hat i +\hat j +\hat k ) \; + λ (\hat i -\hat j + \hat k)\)
= (1 + λ)î + (1 - λ)ĵ + (1 + λ)k̂ .... (1)
Projection of \(\rm \vec v\) on \(\rm \frac {\vec c} {|\vec c|}\) = \(\frac 1 {\sqrt 3}\)
⇒ \(\rm \vec v\). \(\rm \frac {\vec c} {|\vec c|}\) = \(\frac 1 {\sqrt 3}\)
⇒ \(\frac {(1 + λ) - (1 - λ) - (1 + λ)}{\sqrt3} = \frac {1}{\sqrt 3}\)
⇒ -(1 - λ) = 1
∴ λ = 2
Now, put the value of λ in equation (1), we get
\(\rm \vec v\) = 3î - ĵ + 3k̂