Question

Show that the points $O(0,0,0),A(2,-3,3),B(-2,3,-3)$ are collinear. Find the ratio in which each point divides the segments joining the other two.

Answer 1

$\to$ Area of triangled formes by these would be $\frac{1}{2}|\overline{OA}\times \overline{OB}|$

$=\frac{1}{2}|(2\hat{i}-3\hat{j}+3\hat{k})\times (-2\hat{i}+3\hat{j}-3\hat{k})|$

$\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& -3& 3\\ -2& 3& -3\end{array}\right|$

$\hat{i}\left(0\right)+j\left(0\right)+\hat{k}\left(0\right)=0$

$\Rightarrow O,A,B$ are collinear

$\Rightarrow 0=\frac{-2k+2}{k+1}$

$\Rightarrow k=1$

$2=\frac{-2k}{k-1}$

$\Rightarrow -2k=2k+2$

$\Rightarrow k=1/2$

$\Rightarrow -2=\frac{2k}{k-1}$

$\Rightarrow -2k-2=2k$

$\Rightarrow =1/2$