Find the equation of the midparallel line given two parallel lines.

Published on November 18, 2024

Geometry

The equation of the line equidistant from the parallel lines y = (1/3)x - 4 and y = (1/3)x + 1 is y = (1/3)x - 5/3.

Question

Solve using vector calculus.

Here’s how to find the equation of the line midway between

and

The equation

can be rewritten as

. So, a point on line

and its direction vector is

When

, the position vector of

. So, the point

lies on line

The midpoint between the support vectors of

and

(i.e., the points we identified on each line) is:

So the midpoint is

Since

and

are parallel, the line equidistant from them will also have the same direction vector, which is

Using the midpoint

and the direction vector

, the vector equation of the middle parallel is:

From the vector equation, we have

and

. Solving the second equation for

, we get

. Substituting this into the first equation gives:

Final Answer: The final answer is