find-the-angle-between-3y-x-3-1-3-2-y-x-2-help-me-sir-

Question Number 112902 by mohammad17 last updated on 10/Sep/20

f i n d t h e a n g l e b e t w e e n 3 y + \frac{x}{\sqrt{3}} = 1, \frac{\sqrt{3}}{2} y - x = 2

h e l p m e s i r

Answered by bemath last updated on 10/Sep/20

m_{1} = \frac{- \frac{1}{\sqrt{3}}}{3} = - \frac{1}{3 \sqrt{3}}; m_{2} = \frac{1}{(\frac{\sqrt{3}}{2})} = \frac{2}{\sqrt{3}}

say the angle between both line

is α then \tan α =∣ \frac{m_{2} - m_{1}}{1 + m_{2} . m_{1}} ∣

\tan α =∣ \frac{\frac{6 + 1}{3 \sqrt{3}}}{1 - \frac{2}{9}} ∣ = \frac{\frac{21}{\sqrt{3}}}{7} = \frac{3}{\sqrt{3}}

α = \tan^{- 1} (\sqrt{3}) = 60 °

Commented by mohammad17 last updated on 10/Sep/20

t h a n k y o u s i r

Answered by mathmax by abdo last updated on 10/Sep/20

3 y + \frac{x}{\sqrt{3}} = 1 \Rightarrow 3 \sqrt{3} y + x = \sqrt{3} \Rightarrow x + 3 \sqrt{3} y - \sqrt{3} = 0

\Rightarrow the director verctor is \vec{u} (- 3 \sqrt{3}, 1)

\frac{\sqrt{3}}{2} y - x = 2 \Rightarrow \sqrt{3} y - 2 x = 4 \Rightarrow - 2 x + \sqrt{3} y - 4 = 0 \Rightarrow 2 x - \sqrt{3} y + 4 = 0 \Rightarrow

the director vector is \vec{v} (\sqrt{3}, 2)

\cos (u, v) = \frac{u . v}{∣ u ∣ . ∣ v ∣} and \sin (u, v) = \frac{\det (u, v)}{∣ u ∣ . ∣ v ∣} \Rightarrow \tan (u, v) = \frac{\det (u, v)}{u . v}

= \frac{| \begin{matrix} - 3 \sqrt{3} \sqrt{3} \\ 1 2 \end{matrix} |}{(- 3 \sqrt{3}) . \sqrt{3} + 1 \times 2} = \frac{- 7 \sqrt{3}}{- 7} = \sqrt{3}

we have \tan (u, v) = \sqrt{3} \Rightarrow (u, v) = (u, v) = \frac{π}{3} [π]

Commented by mohammad17 last updated on 10/Sep/20

t h a n k y o u s i r

Commented by mathmax by abdo last updated on 11/Sep/20

you are welcome

Answered by Dwaipayan Shikari last updated on 10/Sep/20

3 y + \frac{x}{\sqrt{3}} = 1 \frac{\sqrt{3}}{2} y - x = 2 \Rightarrow y = \frac{2}{\sqrt{3}} x + \frac{4}{\sqrt{3}}

y = - \frac{x}{3 \sqrt{3}} + \frac{1}{3}

t a n θ =∣ \frac{- \frac{1}{3 \sqrt{3}} - \frac{2}{\sqrt{3}}}{1 - \frac{2}{9}} ∣= \sqrt{3}

θ = \frac{π}{3}

Commented by mohammad17 last updated on 10/Sep/20

t h a n k y o u s i r