What are the components of the vector between the origin and the polar coordinate $(4, \frac{11 π}{12})$ $(4, (11pi)/12)$ ?

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What are the components of the vector between the origin and the polar coordinate $(4, \frac{11 π}{12})$ $(4, (11pi)/12)$ ?

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Answer 1 · 2017-09-30T07:14:05

The vector is $= - (\sqrt{2} + \sqrt{6}) ˆ i + (\sqrt{6} - \sqrt{2}) ˆ j$ $=-(sqrt2+sqrt6)hati+(sqrt6-sqrt2)hatj$

Explanation:

If the polar coordinates of the vector are $(r, θ)$ $(r,theta)$ , then

The components are $(r cos θ, r sin θ)$ $(rcostheta,rsintheta)$ in the rectangular cordinates are

$x = r cos θ ˆ i$ $x=rcosthetahati$

$y = r sin θ ˆ j$ $y=rsinthetahatj$

Here, we have

$(r, θ) = (4, \frac{11}{12} π)$ $(r,theta)=(4,11/12pi)$

$x = 4 cos (\frac{11}{12} π) = 4 cos (\frac{2}{3} π + \frac{1}{4} π)$ $x=4cos(11/12pi)=4cos(2/3pi+1/4pi)$

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

$= 4 (- \frac{1}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2})$ $=4(-1/2*sqrt2/2-sqrt3/2*sqrt2/2)$

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

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

$y = 4 sin (\frac{11}{12} π) = 4 sin (\frac{2}{3} π + \frac{1}{4} π)$ $y=4sin(11/12pi)=4sin(2/3pi+1/4pi)$

$= 4 (sin (\frac{2}{3} π) \cdot cos (\frac{1}{4} π) + cos (\frac{2}{3} π) \cdot sin (\frac{1}{4} π))$ $=4(sin(2/3pi)*cos(1/4pi)+cos(2/3pi)*sin(1/4pi))$

$= 4 \cdot (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$ $=4*(sqrt3/2*sqrt2/2-1/2*sqrt2/2)$

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

The vector is

$= - (\sqrt{2} + \sqrt{6}) ˆ i + (\sqrt{6} - \sqrt{2}) ˆ j$ $=-(sqrt2+sqrt6)hati+(sqrt6-sqrt2)hatj$

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What are the components of the vector between the origin and the polar coordinate $(4, \frac{11 π}{12})$ $(4, (11pi)/12)$ ?

1 Answer

Explanation:

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What are the components of the vector between the origin and the polar coordinate (4, (11pi)/12)(4,11π12)(4, (11pi)/12)?

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What are the components of the vector between the origin and the polar coordinate $(4, \frac{11 π}{12})$ $(4, (11pi)/12)$ ?