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.