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(C) Two vectors $\vec{a} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k}$ and $\vec{b} = a_2 \hat{i} + b_2 \hat{j} + c_2 \hat{k}$ are collinear if their com

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$1$

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(C) Two vectors $\vec{a} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k}$ and $\vec{b} = a_2 \hat{i} + b_2 \hat{j} + c_2 \hat{k}$ are collinear if their components are proportional,i.e.,$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = k$ for some constant $k$.Given vectors are $\vec{a} = \hat{i} + 2 \hat{j} + m \hat{k}$ and $\vec{b} = \hat{i} + m \hat{j} + 2 \hat{k}$.For these to be collinear,we must have $\frac{1}{1} = \frac{2}{m} = \frac{m}{2}$.From $\frac{1}{1} = \frac{2}{m}$,we get $m = 2$.From $\frac{1}{1} = \frac{m}{2}$,we get $m = 2$.Since both conditions yield the same value $m = 2$,there is only $1$ such value of $m$ for which the vectors are collinear.

The number of values of $m \in R$ for which the vectors $\hat{i}+2 \hat{j}+m \hat{k}$ and $\hat{i}+m \hat{j}+2 \hat{k}$ are collinear is

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