Question

A vector is perpendicular to both the vectors \(\rm\hat{2i} +\hat{3j}\) and \(\rm \hat{j} +\hat{3k}\) is
1. 9î - 6ĵ + 2k̂  
2. 2î + 9k̂  
3. 2î - 6ĵ
4. 2î - 6ĵ + 3k̂  

Answer 1

Correct Answer - Option 1 : 9î - 6ĵ + 2k̂  

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 2î + 3 ĵ and ĵ + 3k̂

Let \(\rm \vec a \) = 2î + 3ĵ and \(\rm \vec b \) = ĵ + 3k̂

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

= (2î + 3ĵ) × (ĵ + 3k̂)

= 2(î × ĵ) + 6(î × k̂) + 3(ĵ × ĵ) + 9(ĵ × k̂) 

= 2k̂ - 6ĵ + 0 + 9î 

= 9î - 6ĵ + 2k̂