What are the components of the vector between the origin and the polar coordinate $(2, \frac{13 π}{12})$ $(2, (13pi)/12)$ ?

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What are the components of the vector between the origin and the polar coordinate $(2, \frac{13 π}{12})$ $(2, (13pi)/12)$ ?

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Answer 1 · 2017-12-15T08:17:08

The vector is $= < - \frac{(\sqrt{2} + \sqrt{6})}{2}, \frac{(\sqrt{2} - \sqrt{6})}{2} >$ $=<-((sqrt2+sqrt6))/2, ((sqrt2-sqrt6))/2 >$

Explanation:

To convert from polar coordinates $(r, θ)$ $(r, theta)$ to rectangular coordinates , we apply the following equations

$x = r cos θ$ $x=rcostheta$

$y = r sin θ$ $y=rsintheta$

Here,

The polar coordinates are

$r = 2$ $r=2$

and

$θ = \frac{13}{12} π$ $theta=13/12pi$

Therefore,

The rectangular coordinates are

$x = 2 cos (\frac{13}{12} π) = 2 cos (\frac{1}{3} π + \frac{3}{4} π) = 2 (cos (\frac{1}{3} π) cos (\frac{3}{4} π) - sin (\frac{1}{3} π) sin (\frac{3}{4} π))$ $x=2cos(13/12pi)=2cos(1/3pi+3/4pi)=2(cos(1/3pi)cos(3/4pi)-sin(1/3pi)sin(3/4pi))$

$= 2 ((\frac{1}{2}) \cdot (- \frac{\sqrt{2}}{2}) - (\frac{\sqrt{3}}{2}) \cdot (\frac{\sqrt{2}}{2}))$ $= 2((1/2) * (-sqrt2/2)-(sqrt3/2) * (sqrt2/2))$

$= 2 \frac{- \sqrt{2} - \sqrt{6}}{4}$ $=2(-sqrt2-sqrt6)/4$

$= - \frac{\sqrt{2} + \sqrt{6}}{2}$ $=-(sqrt2+sqrt6)/2$

$y = 2 sin (\frac{13}{12} π) = 2 sin (\frac{1}{3} π + \frac{3}{4} π) = 2 (sin (\frac{1}{3} π) cos (\frac{3}{4} π) + cos (\frac{1}{3} π) sin (\frac{3}{4} π))$ $y=2sin(13/12pi)=2sin(1/3pi+3/4pi)=2(sin(1/3pi)cos(3/4pi)+cos(1/3pi)sin(3/4pi))$

$= 2 ((\frac{\sqrt{3}}{2}) \cdot (- \frac{\sqrt{2}}{2}) + (\frac{1}{2}) \cdot (\frac{\sqrt{2}}{2}))$ $= 2((sqrt3/2) * (-sqrt2/2)+(1/2) * (sqrt2/2))$

$= 2 \frac{\sqrt{2} - \sqrt{6}}{4}$ $=2(sqrt2-sqrt6)/4$

$= \frac{\sqrt{2} - \sqrt{6}}{2}$ $=(sqrt2-sqrt6)/2$

Finally,

The vector is $= < - \frac{(\sqrt{2} + \sqrt{6})}{2}, \frac{(\sqrt{2} - \sqrt{6})}{2} >$ $=<-((sqrt2+sqrt6))/2, ((sqrt2-sqrt6))/2 >$

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What are the components of the vector between the origin and the polar coordinate $(2, \frac{13 π}{12})$ $(2, (13pi)/12)$ ?

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Explanation:

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What are the components of the vector between the origin and the polar coordinate (2, (13pi)/12)(2,13π12)(2, (13pi)/12)?

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What are the components of the vector between the origin and the polar coordinate $(2, \frac{13 π}{12})$ $(2, (13pi)/12)$ ?