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# The slope of a line is $-\dfrac{1}{3}$. How do you find the slope of a line that is perpendicular to this line?

Last updated date: 12th Aug 2024
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Hint: Slope at a point of a curve (function) is equal to the tan of the angle that the tangent to the curve at that point makes with positive x-axis. If two lines are perpendicular to each other then the slopes of the two lines are negative reciprocals of each other.

Complete step-by-step solution:
Let us first understand what is meant by slope of a function. Slope at a point of a curve (function) is equal to the tan of the angle that the tangent to the curve at that point makes with positive x-axis.
A line is a curve that has a constant slope or we can say that a line is a curve that has equal slope at all the points.

If two lines are perpendicular to each other then the slopes of the two lines are negative reciprocals of each other. If the slopes of the two lines are ${{m}_{1}}$ and ${{m}_{2}}$ respectively, then the two slopes are related as ${{m}_{1}}{{m}_{2}}=-1$ or ${{m}_{1}}=-\dfrac{1}{{{m}_{2}}}$ ….. (i)
It is given that there are two lines perpendicular to each. One of the lines has a slope of $-\dfrac{1}{3}$ .
This means that ${{m}_{1}}=-\dfrac{1}{3}$.
Then by equation (i) we get that ${{m}_{2}}=-\dfrac{1}{{{m}_{1}}}=-\dfrac{1}{\left( \dfrac{-1}{3} \right)}$
$\Rightarrow {{m}_{2}}=3$
Therefore, the slope of the line that is perpendicular to the line having a slope of $-\dfrac{1}{3}$ is equal to 3.

Note: Note that we two lines are perpendicular to each other, the angle of one of the lines with the positive x-axis must be acute and the angle of the other line with positive x-axis must be obtuse. We can also see in this the given question. When the angle is acute, slope is positive and when the angle is obtuse, the slope is negative. This helps us in checking whether our answer is correct or not.