Exercise 8: We give the two lines (D): \( 2 x-y-40 \) and \( \left(D^{\prime}\right): 2 x+4 y+1=01 \) ) Give a normal vector and a direction vector of each of the two lines (D) and (D'). 2) Show that (D) and (D') are perpendicular to a point I that we will determine its coordinates. 3\( )(D) \) intersects the \( x^{\prime} x \) axis at point \( J \), and (D') intersects the \( y^{\prime} y \) axis at point K. Determine the coordinates of \( I \) and \( K .4 \) ) Calculate the area of the triangle IJK.

Step 1 ：1) Normal vector of (D): \(\begin{pmatrix} 2 \\ -1 \end{pmatrix}\), direction vector of (D): \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\); normal vector of (D'): \(\begin{pmatrix} 2 \\ 4 \end{pmatrix}\), direction vector of (D'): \(\begin{pmatrix} -4 \\ 2 \end{pmatrix}\)

Step 2 ：2) Equating dot products of direction vectors to 0: \((1)(-4)+(2)(2)=0\); Finding point I: intercept form of (D) and (D'): \(y=2x-40\) and \(y=-\frac{1}{2}x-\frac{1}{4}\); solve for I: \(x_I=\frac{39}{5}\), \(y_I=-\frac{63}{5}\)

Step 3 ：3) Coordinates of J: \((20,0)\); Coordinates of K: \(\left( 0, -\frac{1}{4} \right)\)

Step 4 ：4) Area of triangle IJK: \(\frac{1}{2}|(20)(-\frac{63}{5} - -\frac{1}{4}) - (0)(-\frac{63}{5} - 0)| = 120\)