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A vector is perpendicular to both the vectors \(\rm\hat{i} +\hat{j}\) and \(\rm \hat{j} +\hat{k}\) is
1. î - 2ĵ + k̂
2. î + ĵ + k̂
3. î - ĵ + k̂
4. None of these

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Best answer
Correct Answer - Option 3 : î - ĵ + k̂

Concept:

Let \(\rm\vec{a}\) and \(\rm\vec{b}\) be the two vectors and the vector \(\rm\vec{c}\) perpendicular to both \(\rm\vec{a}\) and \(\rm\vec{b}\)

Hence  \(\rm\vec{c} = ​​\rm\vec{a} × \rm\vec{b}\)

 

Calculation:

Let vector \(\rm\vec{c}\) is perpendicular to both the vectors î + ĵ and ĵ + k̂

Let \(\rm \vec a \) = î + ĵ and \(\rm \vec b \) = ĵ + k̂

Therefore, \(\rm\vec{c} = ​​\rm\vec{a} × \rm\vec{b}\)

= (î + ĵ) × (ĵ + k̂)

= î × ĵ + î × k̂ + ĵ × ĵ + ĵ × k̂ 

= k̂ - ĵ + 0 + î 

= î - ĵ + k̂  

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