Let $L_{1}: x-y-2=0$ be a straight line in the x-y coordinate system. Find a coordinate system $(x_{1},y_{1})$ having its origin at $(0,0)$ and relative to which $L_{1}$ has equation $y_{1}= $constant.
I know I keep asking these stupid questions, but my final is tomorrow and I can help feeling like these notes aren't helping me at all. Can somebody explain the steps to me?
$L_1$ has a slope of $1$ so it is at a $45^{\circ}$ angle to both of the current coordinate axes. We know that lines of the form $y=k$ are parallel to the $x$ axis and perpendicular to the $y$ axis so we must rotate our coordinate axes by $45^{\circ}$ in the clockwise direction.
When we do that we get that $y_1$ is the line $y=-x$ in our current coordinate system and $x_1$ is the line $y=x$. Can you show that the equation for $L_1$ in this new coordinate system is $y_1=-\sqrt{2}$?