For the unit vector #hattheta#, geometrically show that #hattheta = -sinthetahati + costhetahatj#? Essentially, converting from cartesian to polar, how would I determine the unit vector for #vectheta# in terms of #theta#, #hati#, and #hatj#?
I've been able to show that #hatr = costhetahati + sinthetahatj# :
#costheta = hati/hatr#
#sintheta = hatj/hatr#
#=> ||hatr|| = sqrt(hatrcdothatr(cos^2theta + sin^2theta))#
#= sqrt(hatrcdothatrcos^2theta + hatrcdothatrsin^2theta)#
#= sqrt(hatrcostheta * hati + hatrsintheta hatj)#
#=> hatrcdothatr = ||hatr||^2 = hatrcosthetacdothati + hatrsinthetacdothatj#
#= hatrcdot(costhetahati + sinthetahatj)#
Thus, #hatr = costhetahati + sinthetahatj# . But how would I do it for #hattheta# ? I'm probably just missing something really simple, like where the #hattheta# vector points.
I've been able to show that
Thus,
See the design in the explanation and the graph.
The parallel position vector through the origin O ( r = 0 ) is
graph{(x^2+y^2-1)(y-x/sqrt3)(y+sqrt3x)=0 [-1, 1, -.05, 1]}
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See below.
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To determine the unit vector (\hat{\theta}) in terms of (\theta), (\hat{i}), and (\hat{j}) in Cartesian coordinates, you can use the trigonometric relationships between Cartesian coordinates and polar coordinates.
The unit vector (\hat{\theta}) is perpendicular to the radial direction and points in the direction of increasing polar angle (\theta). In Cartesian coordinates, it can be expressed as (\hat{\theta} = -\sin(\theta)\hat{i} + \cos(\theta)\hat{j}).
This expression arises from the fact that in polar coordinates, the unit vector (\hat{\theta}) can be represented by the sine and cosine of the angle (\theta), with (\hat{i}) and (\hat{j}) representing the unit vectors along the x and y directions respectively.
So, (\hat{\theta} = -\sin(\theta)\hat{i} + \cos(\theta)\hat{j}) represents the unit vector (\hat{\theta}) in terms of Cartesian unit vectors (\hat{i}) and (\hat{j}) and the angle (\theta) in polar coordinates.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
- What is the polar form of #( 104,105 )#?
- What is the slope of the polar curve #f(theta) = 2theta + tan^2theta - costheta # at #theta = (pi)/8#?
- What is the equation of the tangent line to the polar curve # f(theta)= 5thetasin(3theta)+2cot(11theta) # at #theta = pi/12#?
- What is the distance between the following polar coordinates?: # (-4,(3pi)/4), (5,(3pi)/8) #
- What is the distance between the following polar coordinates?: # (3,(pi)/4), (9,(3pi)/8) #

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