Write the equation (in coordinates $x_1, x_2)$ of a line in $A^2$, where A denotes Affine space: passing through the point (2, −3) and parallel to the vector (5, 2).
Can we use the equation y=mx +c to find the slope and the constant to write the equation in Affine space?
Like I can get the slope as m = 2/5 since its parallel to the vector (5,2) and can get the 'c' by substituting the value of the point(2,-3) and m in the above-mentioned equation. I get the equation as $y=(2/5)x-(19/5)$.
In general, if the line passes through $P=(p_1, p_2)$ and is parallel to the vector $V = (v_1, v_2)$, one can say that $X=(x_1, x_2)$ belongs to the line if the vector $\overrightarrow{PX}$ is colinear to $V$, that is to say if the determinant \begin{equation} 0= \begin{vmatrix} x_1-p_1& v_1\cr x_2 -p_2& v_2\cr \end{vmatrix} = v_2(x_1-p_1)-v_1(x_2-p_2) \end{equation} This is the equation of the line. In your case \begin{equation} 2(x_1-2) - 5(x_2+3) = 0 \end{equation} or $2 x_1-5 x_2=19$.