The point of intersection of the lines joining points i ^ + 2 j ^ ,   2 i ^ - j ^ and - i ^ ,   2…

Given that: $\hat{i}+2\hat{j}, 2\hat{i}-\hat{j} \& -\hat{i}, 2\hat{i}$

$A\left(1,2\right), B\left(2,-1\right), C\left(-1,0\right), D\left(2,0\right)$

Equation of line passing through two points is $\left(y-y_1\right)=\left(\frac{y_2-y_1}{x_2-x_1}\right)\left(x-x_1\right)$

Equation of line though $AB: \left(y-2\right)=\left(\frac{-1-2}{2-1}\right)\left(x-1\right) \Rightarrow 3x+y=5$

Equation of line through $CD: \left(y-0\right)=\left(\frac{0-0}{2+1}\right)\left(x+1\right) \Rightarrow y=0$

Solving the above equations, we get

$3x+0=5$

$\Rightarrow x=\frac{5}{3}, y=0$

Hence, the point of intersection is $\frac{5}{3}\hat{i}$.