Let \(\vec{a} =\hat{\imath} + 2\hat{\jmath}+3\hat{k}, \vec{b} = -\hat{\imath}+ 2\hat{\jmath} +\hat{k},\vec{c} = 3\hat{\imath}+\hat{\jmath}\) be three vectors. If \((\vec{a}+\lambda\vec{b})\) is perpendicular to \(\vec{c}\), then the value of \(\lambda\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 5
Adding component by component:
\(\vec{a} + \lambda\vec{b} = (1 - \lambda)\hat{\imath} + (2 + 2\lambda)\hat{\jmath} + (3 + \lambda)\hat{k}\)
Two vectors are perpendicular when their dot product is zero. With \(\vec{c} = 3\hat{\imath} + \hat{\jmath} + 0\hat{k}\):
\((\vec{a} + \lambda\vec{b})\cdot\vec{c} = 3(1 - \lambda) + 1(2 + 2\lambda) + 0(3 + \lambda)\)
\(= 5 - \lambda\)
\(5 - \lambda = 0 \Rightarrow \lambda = 5\)