GATE (TF) Textile 2018 Question Paper Solution | GATE/2018/TF/20

Question 20 (Textile Technology & Fibre Science)

If \vec{a}=-\hat{i}+2\hat{j}, \vec{b}=-\hat{j}+2\hat{k} and \vec{c}=2\hat{i}-\hat{k} are three vectors such that \vec{a}+\lambda\vec{b} is perpendicular to \vec{c} then the value of \lambda is __-1 to 1__.

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\vec{a}+\lambda\vec{b}=-\hat{i}+2\hat{j}+\lambda \times (-\hat{j}+2\hat{k})=-\hat{i}+(2-\lambda)\hat{j}+2\hat{k}
And this vector is perpendicular to \vec{c}.
(\vec{a}+\lambda\vec{b}) \times \frac{\vec{c}}{\left | \vec c \right |}=0
(-\hat{i}+(2-\lambda)\hat{j}+2\hat{k}) \cdot \frac{(2\hat{i}-\hat{k})}{\left | (2\hat{i}-\hat{k})\right |} =0
(-\hat{i}+(2-\lambda)\hat{j}+2\hat{k}) \cdot \frac{(2\hat{i}-\hat{k})}{\left | (\sqrt 5\right |} =0

\lambda=-1 (Ans)

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