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# If the point $(\lambda , 0 , 3 ), (1 , 3 , -1 )$ and $(-5 , -3 , 7 )$ are collinear than find $\lambda$.

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• If the points $A(a_1, a_2, a_3 ), B(b_1, b_2, b_3), C(c_1, c_2, c_3)$ are collinear, then their position vectors are collinear. Then $[ \overrightarrow a\: \overrightarrow b\: \overrightarrow c] =\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}=0$
Since $A(\lambda, 0, 3), B(1, 3, -1), C(-5, -3, 7)$ are collinear $[\overrightarrow a\: \overrightarrow b\: \overrightarrow c]=0$
$\therefore \begin{vmatrix} \lambda & 0 & 3 \\ 1 & 3 & -1 \\ -5 & -3 & 7 \end{vmatrix}$
$= \lambda(21-3)+3(-3+15)=0$
$\Rightarrow 18\lambda=-36$
$\lambda=-2$

answered Jun 14, 2013
edited Jun 24, 2013