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You are watching: Represent the plane curve by a vector-valued function


Represent the airplane curve by a vector-valued function. (There are plenty of correct answers.)x^2 + y^2 = 25My Work:Let x = tI decided to fix for y simply because this is what concerns mind as action 1.y^2 = 25 - t^2rooty^2 = root25 - r^2My Answer:r(t) = root25 - t^2iBook"s Answer:r(t) = 5 price i + 5 sint jWhere do the trig functions come from?Why is my answer wrong?
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this is among those points you need to learn to recognize.$x^2 + y^2 = 25$ is a circle of radius 5. The most natural method to represent a one is polar coordinates.your vector is $5 cos( heta), 5sin( heta),~0 leq heta or ns guess they usage the unit vectors to attain $5cos( heta) hati + 5 sin( heta) hatj,~0 leq heta note you don"t have to restrict $ heta$ however you"ll end up repeating the same circle.note that your prize a) isn"t a vectorb) if friend had gotten it ideal as a vector, i.e. $ , sqrt25-t^2$, it would just cover half the circleyou would have to incorporate it"s enjoy $ , -sqrt25-t^2$ to obtain the entire circle.
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this is just one of those points you need to learn to recognize.$x^2 + y^2 = 25$ is a one of radius 5. The many natural way to stand for a one is polar coordinates.your vector is $5 cos( heta), 5sin( heta),~0 leq heta or ns guess they usage the unit vectors to achieve $5cos( heta) hati + 5 sin( heta) hatj,~0 leq heta note you don"t need to restrict $ heta$ however you"ll end up repeating the very same circle.note the your price a) isn"t a vectorb) if friend had obtained it best as a vector, i.e. $ , sqrt25-t^2$, that would just cover half the circleyou would have actually to incorporate it"s reflection $ , -sqrt25-t^2$ to gain the whole circle.
This one entails thinking outside the box. Ns quickly construed the offered equation to be a circle of radius 5 however was no too certain where the trig attributes came from.
The equation that a one with facility at the origin is (displaystyle x^2+ y^2= R^2). The truth that (displaystyle cos^2(t)+ sin^2(t)= 1) leader to (displaystyle R^2cos^2(t)+ R^2sin^2(t)= R^2) and makes the parametric equations (displaystyle x= R cos(t)), (displaystyle y= R sin(t)) quite natural.

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The equation the a circle with facility at the beginning is (displaystyle x^2+ y^2= R^2). The reality that (displaystyle cos^2(t)+ sin^2(t)= 1) leads to (displaystyle R^2cos^2(t)+ R^2sin^2(t)= R^2) and also makes the parametric equations (displaystyle x= R cos(t)), (displaystyle y= R sin(t)) nice natural.