How do you use trigonometric formulas (or identities) to prove that the product of the gradients of two perpendicular lines is -1?
If $y = m_1x + c_1 \text{ and } y = m_2x + c_2,$
I thought finding an angle would help to incorporate one of the identities, and hence get somewhere. But how do I find an angle? Constructing two vectors in terms of the y-intercepts and x-intercepts of the two equations like below?
($y_1$ is the y-intercept of the first equation, $x_1$ is the x-intercept of the first equation, and so on.)
$ \rightarrow \left(\begin{array}{c}x_1\\ y_1\end{array}\right) \text{•} \left (\begin{array}{c}x_2\\ y_2\end{array}\right) \\~\\ \rightarrow \left(\sqrt{(x_1^2 + y_1^2)(x_2^2 + y_2^2)}\hspace{0.08cm} \right) \cos \theta = x_1x_2 + y_1y_2 \\ \theta = \cos^{{-}1} \left[\frac{x_1x_2 + y_1y_2}{\sqrt{(x_1^2 + y_1^2)(x_2^2 + y_2^2)}} \right] $
But this is taking me nowhere!
Thanks in advance.
If $\theta$ is the angle made by the first line with $x-$ axis then the slope of this line is $\tan (\theta)$. The slope of the second line is $\tan (\pi /2+\theta)=-\cot(\theta)$. Since $\tan (\theta)$ $(-\cot (\theta))=-1$ we are done.