Question

If the line segment joining the points $A(a,b)$ and $B(c,d)$ subtends an angle $\theta$ at the origin,then $\cos \theta =ac+bd\sqrt{(a^2+b^2)(c^2+d^2)}$

Answer 1

If$a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$
are two times, the angle between them can be calculated using the following formula.
$\cos \theta =\frac{a_1{a}_2+b_1{b}_2}{\sqrt{({a_1}^2+{b_1}^2)\left({a_2}^2{+b_2}^2\right)}}$
Slope of line $OA=\frac{b}{a}$
lin$OA:(y-0)=\frac{b}{a}(x-0)\Rightarrow bx-ay=0$
Similarly lin$OB:dx-cy=0$
$\therefore \cos \theta =\frac{bd+(-a)(-c)}{\sqrt{(a^2+b^2)\left(c^2{+d}^2\right)}}=\frac{bd+ac}{\sqrt{(a^2+b^2)\left(c^2{+d}^2\right)}}$