Question

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

Answer 1

The correct answer is: 8
The given vectors are $\vec{a}=2\hat{i}+2\hat{j}+3\hat{k}$,$b=-\hat{i}+2\hat{j}+\hat{k}$,and $c=3\hat{i}+\hat{j}$
Now,
$\vec{a}+λ\vec{b}$
=(2\hat{i}+2\hat{j}+3\hat{k})$$+\lambda(-\hat{i}+2\hat{j}+\hat{k})
$=(2-λ)\hat{i}+((2+2λ)\hat{j}+(3+λ)\hat{k}$
if $(\vec{a}+λ\vec{b})$ is perpendicular to $\vec{c}$, then
$(\vec{a}+λ\vec{b}).\vec{c}=0$
$⇒[(2-λ)\hat{i}+(2+2λ)\hat{j}+(3+λ)\hat{k}].(3\hat{i}+\hat{j})=0$
$⇒(2-λ)3+(2+2λ)1+(3+λ)0=0$
$⇒6-3λ+2+2λ=0$
$⇒-λ+8=0$
$⇒λ=8$
Hence,the required value of $λ$ is 8.