find angle between two lines between $y-\sqrt{3}x-5=0$ and $\sqrt{3}y-x+6$

Question

how to find to angle between these two lines $y-\sqrt{3}x-5=0$ and $\sqrt{3}y-x+6=0$

i tried so far like this
$y-\sqrt{3}x-5=0$

$y=\sqrt{3}x+5$

in the form of $y=mx+b$

got the value for $m_1=\sqrt{3}$

and for $\sqrt {3}y-x+6=0$

$y=\dfrac {x-6} {\sqrt {3}}$

$y=\dfrac {1} {\sqrt {3}}x-6$

$m_{2}=\dfrac {1} {\sqrt {3}}$

the formula to find angle between two lines

$\tan \theta =\dfrac {m_{2-}m_{1}} {1+m_{1}m_{2}}$

$\frac{\frac{1}{\sqrt{3}}-\sqrt{3}}{1+\sqrt{3}.\frac{1}{\sqrt{3}}}$

$\frac { - \frac { 2 } { \sqrt { 3 } } } { 1 + 1 }$

$\frac { - \frac { 2 } { \sqrt { 3 } } } { 2 }$

$- \frac { 2 } { 2 \sqrt { 3 } }$

$- \frac { 1 } { \sqrt { 3 } }$

is this right till now and how to find angle after wards . should i use tan inverse of $- \frac { 1 } { \sqrt { 3 } }$ or my algebra calculation is wrong . help me thank you

Answer 1

Recall that to find the angle we can refer to the lines parallel to the given and passing through the origin, that is

• $y=\sqrt{3}x$

• $\sqrt{3}y=x$

then, using parametric form, the direction vectors are

• for $y=\sqrt{3}x$ $$v_1=(1,\sqrt 3)$$

• for $\sqrt{3}y=x$ $$v_2=(\sqrt 3,1)$$

finally we can compute the angle by dot product

$$\cos \theta = \frac{v_1\cdot v_2}{\|v_1\|\|v_2\|}=\frac{2\sqrt 3}{4}=\frac{\sqrt 3}{2}\implies \theta = 30°$$

As an alternative the angles between the two lines and the $x$ axis are

• for $y=\sqrt{3}x$ $$\tan \theta_1=\sqrt 3 \implies \theta_1=60°$$

• for $\sqrt{3}y=x$ $$\tan \theta_2=\frac1{\sqrt 3} \implies \theta_2=30°$$

$$\implies \theta =\theta_1-\theta_2= 30°$$

Then the acute angle between the two lines is equal to $30°=\frac{\pi}6$ and as a consequence the obtuse angle between the two lines is equal to $180°-30°=150°=\frac{5\pi}6$.

Answer 2

Your calculation is very good so far.

You have found the tangent of the angle between two line to be $$ \tan \theta = \frac {-1}{\sqrt 3}$$

Thus $$ \theta = \tan ^{-1} \frac {-1}{\sqrt 3}=-\pi /6$$

You want a positive value for your angle so you add $\pi$ to get $$ \pi -\pi/6 = 5\pi/6$$ which is $150$ degrees.

Answer 3

It's a very simple problem in essence. The gradients of the two lines are $\sqrt 3$ and $\frac 1{\sqrt 3}$. Recall that gradients equal the tangents of the angles made by lines with the $x$-axis. The special angles that give those tangents can quickly be recognised to be $60^{\circ}$ and $30^{\circ}$ (both in the first quadrant), giving the difference as $30^{\circ}$.