Question

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

Answer 1

The given vectors are $a=2\hat{i}+2\hat{j}+3\hat{k}$, $b=-\hat{i}+2\hat{j}+\hat{k}$ and $c=3\hat{i}+\hat{j}$
Now, $(a+\lambda b)\perp c$ (Given)
$\Rightarrow (a+\lambda b).c=0$
($\because$ scalar product of two perpendicular vectors is zero)
$\Rightarrow \left[\right(2\hat{i}+2\hat{j}+3\hat{k})+\lambda (-\hat{i}+2\hat{j}+\hat{k}\left)\right].(3\hat{i}+\hat{j})=0\Rightarrow \left[\right(2-\lambda )\hat{i}+(2+2\lambda )\hat{j}+(3+\lambda \left)\hat{k}\right].(3\hat{i}+\hat{j})=0\Rightarrow (2-\lambda )3+(2+2\lambda )1+(3+\lambda )0=0\Rightarrow 6-3\lambda +2+2\lambda =0\Rightarrow 8-\lambda =0\Rightarrow \lambda =8$
Hence, the required value of $\lambda$ is 8.