Question

What is the slope of the line that is perpendicular to a slope of $-\dfrac{1}{6}?$

Answer 1

We use the concept of perpendicular lines to calculate the slope of the line perpendicular to the slope of the given line. It is known that for two lines with slopes ${{m}_{1}}$ and ${{m}_{2}}$ , the condition for them to be perpendicular is given by ${{m}_{1}}.{{m}_{2}}=-1.$ Using the slope given, we calculate the slope of the other line using the above condition.

Complete step by step solution:
In order to solve this question, we need to show the condition for perpendicular lines. Two lines are said to be perpendicular if they satisfy the equation,
$\Rightarrow {{m}_{1}}.{{m}_{2}}=-1$
Where ${{m}_{1}}$ is the slope of the first line and ${{m}_{2}}$ is the slope of the second line.
Given the slope of the line as $-\dfrac{1}{6},$ let us assume this to be the slope of the first line, that is ${{m}_{1}}=-\dfrac{1}{6}.$
We are to find the slope of the line perpendicular to the given line. In order to do this, we substitute the value of ${{m}_{1}}$ in the above condition and obtain the slope of the line perpendicular to this one.
$\Rightarrow -\dfrac{1}{6}.{{m}_{2}}=-1$
Multiplying both sides with -6,
$\Rightarrow {{m}_{2}}=-1\times -6$
Taking a product of the two terms on the right-hand side,
$\Rightarrow {{m}_{2}}=6$
Hence, the slope of the line that is perpendicular to a slope of $-\dfrac{1}{6}$ is 6.

Note: We need to know the slope of perpendicular lines and the condition to show two lines are perpendicular in order to solve such questions. It is important to note that the slope of a line is given by ${{m}_{1}}$ which can be calculated as ${{m}_{1}}=\tan \theta ,$ where $\theta$ is the angle that the line makes with the positive direction of the x-axis.