Q.Find the angle between the two lines whose slopes are $2$ and $-\dfrac13$, and determine whether the lines are perpendicular.

With both slopes given directly, the angle formula applies at once; perpendicularity is then checked separately using the product-of-slopes condition.

The angle $\theta$ between two lines of slopes $m_1$ and $m_2$ satisfies:

Here $m_1=2$, $m_2=-\dfrac13$.

Numerator: $m_1-m_2 = 2-\left(-\dfrac13\right) = 2+\dfrac13 = \dfrac73$.

Denominator: $1+m_1m_2 = 1+2\left(-\dfrac13\right) = 1-\dfrac23 = \dfrac13$.

So $\tan\theta = \left|\dfrac{7/3}{1/3}\right| = |7| = 7$, giving $\theta = \arctan(7) \approx 81.87°$.

Perpendicularity check. Two lines are perpendicular exactly when $m_1 m_2 = -1$. Here $m_1 m_2 = 2 \times \left(-\dfrac13\right) = -\dfrac23$, which is not $-1$, so the lines are not perpendicular. …

Confusing the perpendicularity condition ($m_1m_2=-1$) with the parallel condition ($m_1=m_2$) is a common mix-up; also, forgetting the absolute value in the angle formula can produc …