I'm just working on some summer problems so that I can be more prepared when I go into my class in the fall. I found a website full of problems of the content we will be learning but it doesn't have the answers. I need a little guidance on how to do this problem. Here is the problem:
Consider the lines $L1$ and $L2$ with equations:
$L_1 : r = (11, 8, 2) + s(4, 3, -1)$
$L_2 : r = (1, 1,-7) + t(2, 1, 11)$
The lines intersect at point $P$.
The coordinates of $P$ are (3, 2, 4).
The lines are also perpendicular.
The point $Q(7,5,3)$ lies on $L_1$. The point $R$ is the reflection of $Q$ in the line $L_2$.
Question: Find the coordinates of $R$.
So, trying to work this out, Q is on $L_1$. $L_1 : r = (11, 8, 2) + s(4, 3, -1)$. And $R$ is on $L_2$. $L_2 : r = (1, 1,-7) + t(2, 1, 11)$
I’m guessing I figure out how $Q$ is correlated to $L_1$. And then relate that to $R$. But how would that work?
Best Answer
The reflection of a point $Q$ in a line $L$ can be described as follows:
Drop a perpendicular from $Q$ to the line $L.$ Suppose we say $M$ is the name of the point at the foot of the perpendicular. That is, $M$ is on $L$ and the segment $QM$ is perpendicular to $L.$
The reflection of $Q$ in the line $L$ is the point (call it $Q'$) that is the same distance from $M$ as $Q$ is, but in the exact opposite direction. That way, the segment $QQ'$ is perpendicular to $L,$ and $L$ cuts through $QQ'$ at $M,$ exactly midway between $Q$ and $Q'$.
One way to look at this is, if you have $M,$ you can take the vector from $M$ to $Q,$ and then reverse the direction of the vector to get a vector from $M$ to $Q'.$ Now, knowing $M$ and the vector from $M$ to $Q',$ you find $Q'.$
Now review the information in the question statement. You have the coordinates of $Q.$ You also have the coordinates of $M$ (not called by that name, but if you look at the facts already given in the question, you should find the coordinates there). So you can find the vector $MQ,$ reverse it, and find $Q'.$
Alternatively, instead of using vectors, use the fact that $M$ is the midpoint of $Q$ and $Q'.$ Therefore $$ x_M = \frac{x_Q + x_{Q'}}{2}, \quad y_M = \frac{y_Q + y_{Q'}}{2}, \quad \text{and}\quad z_M = \frac{z_Q + z_{Q'}}{2}.$$ Put the coordinates of $Q$ and $M$ in these equations, then solve for $x_{Q'},$ $y_{Q'},$ and $z_{Q'}.$