The vectors \(\rm \vec a, \vec b\ and\ \vec c\) are of the same length. If taken pairwise they form equal angles. If \(\rm \vec a=̂ i+̂ j\ and \ \vec b=̂ j+̂ k,\) then what can \(\vec c\) be equal to? 

I. î + k̂ 

II. \(\rm \frac{-\hat i+4\hat j-\hat k}{3}\)

Select the correct answer using the code given below. 

Explanation:

Given:

\(\rm \vec a=̂ i+̂ j\ and \ \vec b=̂ j+̂ k,\)

Also, \(\vec a, \vec b, \vec c\) has same length

⇒ \(|\vec a| = |\vec b| = |\vec c|\) = √2 

Let θ be the angle between the vectors.

⇒ Cosθ = \(\frac{\vec a. \vec b}{|\vec a||\vec b|} = \frac{0+1+0}{\sqrt2} =\frac{1}{\sqrt2}\)

(I) Let \(\vec c =\hat i +\hat k\)

\(|\vec c| = \sqrt2\)

Cosθ = \(\frac{\vec a. \vec c}{|\vec a||\vec c|} = \frac{0+1+0}{\sqrt2} =\frac{1}{\sqrt2}\)

All the conditions are satisfied, so it can be vector \(\vec c\)

(II) Let \(\vec c =\rm \frac{-\hat i+4\hat j-\hat k}{3}\)

⇒ \(|\vec c| = \frac{1}{3}\sqrt18 = \sqrt2\)

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

= \(\frac{\frac{-1}{3}\frac{4}{3}}{\sqrt2 \sqrt2} = \frac{1}{2}\)

All the conditions are satisfied, so it can be vector \(\vec c\)

∴ Option (c) is correct.