Parallel and Perpendicular Lines
Parallel Lines
Parallel lines are two or more lines that are always the same distance apart and never meet, no matter how far they are extended. In other words, they have the same slope and will never intersect. For example, the tracks of a railway are parallel lines.
If the slope coefficients of two lines are known and equal to k1 and k2, then the parallelism condition for these lines has the form
If two lines have the general equations
then their slopes k1 and k2 are respectively
For these lines to be parallel, their slopes must be equal, which means that:
This can be rearranged to give:
So, if the coefficients of x and y in the general equations of two lines satisfy the above condition, then the lines are parallel.
Perpendicular Lines
Perpendicular lines are lines that intersect at a right angle (90 degrees). They have slopes that are negative reciprocals of each other: For example, a vertical line and a horizontal line intersect at right angles and are therefore perpendicular.
or equivalently
For example, the sides of a square are perpendicular to each other.
Another example is the x-axis and the y-axis: The x-axis and the y-axis are two lines that intersect at a right angle (90 degrees) and divide the coordinate plane into four quadrants.
Two straight lines
are perpendicular if
Solved Problems
Example 1.
Show that the lines \({2x - 3y + 4 = 0}\) and \({6x - 9y - 2 = 0}\) are parallel.
Solution.
Convert the given equations into slope-intercept form:
So from here we get
Hence, the lines are parallel.
Example 2.
Show that the lines \({3x - 5y + 1 = 0}\) and \({10x + 6y - 7 = 0}\) are perpendicular.
Solution.
Rewrite the equations of the lines in slope-intercept form:
Here we have
Since
the straight lines are perpendicular.
Example 3.
Find the equation of the line passing through the point of intersection of the lines \({2x - 3y - 1 = 0}\) and \({3x - y - 2 = 0}\) and perpendicular to the line \({x - y + 1 = 0}.\)
Solution.
To find the point of intersection of the lines, we solve the system of equations
Determine the \(x-\)coordinate of the intersection point:
Then the \(y-\)coordinate is
that is, the intersection point has coordinates
Our line must be perpendicular to the line \({x - y + 1 = 0}\) which has the slope \(k_1 = 1.\) Determine the slope \(k_2\) of our straight line from the condition of perpendicularity:
Now we can easily write the equation of the line using the point-slope form:
Therefore
Example 4.
Find the equation of the line passing through the point \(A\left({-3,4}\right)\) and parallel to the line \[{\frac{x-1}{2} = \frac{y+2}{3}}.\]
Solution.
Remember that if a line is defined in the point-direction form
then the vector \(\mathbf{s}\left(X, Y\right)\) is directed along the straight line, and the point \(\left(x_0, y_0\right)\) lies on the line.
It is clear that a line parallel to
and passing through another point \(A\left(x_1, y_1\right) = A\left(-3, 4\right)\) will have the equation
Example 5.
Find the equation of the line passing through the point \(A\left({-3,4}\right)\) and perpendicular to the line \[{x = t + 3,\; y = -7t + 4.}\]
Solution.
The straight line is given by a parametric equation. Let's calculate its slope:
So the slope is \(k = -7.\) Using the point-slope form, we get
Example 6.
At what value of \(a\) the lines \(ax - 4y = 6\) and \(x - ay = 3\) are parallel?
Solution.
Rearrange the given equations and write them in slope-intercept form:
The straight lines are parallel when
Check each value of \(a.\) When \(a = 2:\)
that is, in this case, the two lines coincide.
Now substitute the value \(a = -2:\)
Therefore, we have two different parallel lines at \(a = -2.\)