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 $\vec{c}$ .
Now,
$(\vec{a}+\lambda\vec{b}) \times \frac{\vec{c}}{\left | \vec c \right |}=0$
$(-\hat{i}+(2-\lambda)\hat{j}+2\hat{k}) \cdot \frac{(2\hat{i}-\hat{k})}{\left | (2\hat{i}-\hat{k})\right |} =0$
$(-\hat{i}+(2-\lambda)\hat{j}+2\hat{k}) \cdot \frac{(2\hat{i}-\hat{k})}{\left | (\sqrt 5\right |} =0$

$(2+0+2\lambda)=0$
$\lambda$ =-1 (Ans)

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