Q

# Confused! kindly explain, - The sum of the distinct real values of, for which the vectors, are co-planar, is : - Vector Algebra - JEE Main

The sum of the distinct real values of $\mu$, for which the vectors, $\mu \hat{i}+\hat{j}+\hat{k},\: \: \hat{i}+\mu \hat{j}+\hat{k},\: \: \hat{i}+\hat{j}+\mu \hat{k}$  are  co-planar, is :

• Option 1)

$0$

• Option 2)

$1$

• Option 3)

$2$

• Option 4)

$-1$

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Coplanar vectors -

$x\hat{a}+y\hat{b}+z\hat{c}=0$

- wherein

$\hat{a},\hat{b},\hat{c}$ are coplanar and $x,y,z$ are scalars (not all zero)

$D = \begin{vmatrix} \mu &1 & 1\\ 1 & \mu & 1\\ 1 & 1 & \mu \end{vmatrix} \\\\\ R_1 \rightarrow R_1 + R_2 + R_3\\\\ \Rightarrow D = (\mu + 2)\begin{vmatrix} 1 &1 & 1\\ 1 & \mu & 1\\ 1 & 1 & \mu \end{vmatrix} \\\\ C_3 \Rightarrow C_3 - C_1 \;\;\;\;\;\; C_2 \Rightarrow C_2 - C_1 \\\\ \Rightarrow D = (\mu + 2)\begin{vmatrix} 1 &0 & 0\\ 1 & \mu -1 & 0\\ 1 & 0 & \mu -1\end{vmatrix} \\\\\ \Rightarrow D = (\mu + 2)(\mu -1)^2 = 0 \\\\\Rightarrow \mu = -2 \;\textup{or} \;1$

Sum of real numbers of $\mu = -2 + 1 = -1$

Option 1)

$0$

Option 2)

$1$

Option 3)

$2$

Option 4)

$-1$

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