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 segment joining the other two.​

Answer 1

Given:

O(0,0,0),A(2,-3,3),B(-2,3,-3)are collinear

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

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

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

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

$\text { O, A, B are collinear }$

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

$\begin{array}{l}\Rightarrow \mathrm{k}=1 \\2=\frac{-2 \mathrm{k}}{\mathrm{k}-1} \\\Rightarrow-2 \mathrm{k}=2 \mathrm{k}+2 \\\Rightarrow \mathrm{k}=1 / 2 \\\Rightarrow-2=\frac{2 \mathrm{k}}{\mathrm{k}-1} \\\Rightarrow-2 \mathrm{k}-2=2 \mathrm{k} \\\Rightarrow=1 / 2\end{array}$

Hence the ratio is 1:2