Let a, b and c be distinct positive numbers.


Let $a, b$ and $c$ be distinct positive numbers. If the vectors $a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}$ and $c \hat{i}+c \hat{j}+b \hat{k}$ are co-planar, then $\mathrm{c}$ is equal to:

  1. $\frac{2}{\frac{1}{a}+\frac{1}{b}}$

  2. $\frac{\mathrm{a}+\mathrm{b}}{2}$

  3. $\frac{1}{a}+\frac{1}{b}$

  4. $\sqrt{\mathrm{ab}}$

Correct Option: , 4


Because vectors are coplanar

Hence $\left|\begin{array}{lll}\mathrm{a} & \mathrm{a} & \mathrm{c} \\ 1 & 0 & 1 \\ \mathrm{c} & \mathrm{c} & \mathrm{b}\end{array}\right|=0$

$\Rightarrow \mathrm{c}^{2}=\mathrm{ab} \Rightarrow \mathrm{c}=\sqrt{\mathrm{ab}}$

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