Question

If $\overrightarrow{a}=2\hat{i}+2\hat{j}+3\hat{k},$ $\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}$ and $\overrightarrow{c}=3\hat{i}+\hat{j}$ are such that $\overrightarrow{a}+\lambda \overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$ then find the value of $\lambda$.

Answer 1

Given : $a=2i+2j+3k$ , $b=-i+2j+k$ and $c=3i+j$

$a+\lambda b=2i+2j+3k+\lambda (-i+2j+k)=i(2-\lambda )+j(2+2\lambda )+k(3+\lambda )$

Given that $a+\lambda b$ is perpendicular to $c$

So we have $(a+\lambda b)\cdot c=0$

$⟹\left(i\right(2-\lambda )+j(2+2\lambda )+k(3+\lambda \left)\right).(3i+j)=0$

$⟹3(2-\lambda )+2+2\lambda =0$

$⟹8-\lambda =0$

$⟹\lambda =8$