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Finding the mirror image of a line
Published on November 18, 2024
Geometry
The solution finds the equation of a line g' that is the reflection of line g across line s. It involves finding the intersection point of g and s, projecting a direction vector of g onto a normal vector of s, and then calculating the direction vector of g'. Finally, the equation of g' is determined using the intersection point and the direction vector of g'.
Question
Mirror the straight line g on the straight line s and determine the coordinates of the mirror line g'. g: 2x+y-12=0, s: 2x+3y-12=0
Solve in terms of vectors.
Answer
Here’s how to solve this problem using vectors:
1. Find the intersection point of the two lines:
To find the intersection point, we solve the system of equations:
Subtracting the first equation from the second gives
2. Find a direction vector for line g:
We can rewrite the equation of line g in the form
3. Find a normal vector for line s:
The coefficients of
4. Find the projection of
The projection of
5. Find the direction vector of the reflected line g’:
The direction vector of the reflected line,
6. Determine the equation of the mirrored line g’:
We have a point
Eliminating
Final Answer: The final answer is