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math_proof
 3 years ago
is this vector field F tangent to or normal to the curve C
math_proof
 3 years ago
is this vector field F tangent to or normal to the curve C

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math_proof
 3 years ago
Best ResponseYou've already chosen the best response.0F=<y,x> where C={(x,y):x^2+y^2=1} and n=<x,y>

math_proof
 3 years ago
Best ResponseYou've already chosen the best response.0i think is tangent but i'm not sure

eseidl
 3 years ago
Best ResponseYou've already chosen the best response.1Take the dot product of the normal vector and F: \[<y, x>*<x,y>=xyxy=0\]Since the vectors are orthogonal, the vector field cannot be normal to the curve (if it was, F and n would be parallel). Since your only other choice is parallel, they are parallel.

math_proof
 3 years ago
Best ResponseYou've already chosen the best response.0so its normal at all points to C?

eseidl
 3 years ago
Best ResponseYou've already chosen the best response.1Another way to see this is to note that C is just a unit circle:dw:1353383917635:dwNo, F is tangent to all point of C

math_proof
 3 years ago
Best ResponseYou've already chosen the best response.0yea C is only circle and the vector F goes in circles so that means its tangent at all points on C?

eseidl
 3 years ago
Best ResponseYou've already chosen the best response.1F is tangent because it is orthogonal to the normal vector n.

math_proof
 3 years ago
Best ResponseYou've already chosen the best response.0what if F=<Y,x> and n=<x,y>

eseidl
 3 years ago
Best ResponseYou've already chosen the best response.1The only way F can be perpendicular to n is if F is tangent to the circle. We showed they are perpendicular because their dot product is zero. Thus F is tangent to C.

eseidl
 3 years ago
Best ResponseYou've already chosen the best response.1The F and n would not be orthogonal since their dot product=2xy. dw:1353384419533:dwThey would look something like this

eseidl
 3 years ago
Best ResponseYou've already chosen the best response.1However, when either x=0 or y=0 the dot product is zero and so n and F would be perpendicular at those points (F would be tangent to C at these points)...but in general 2xy doesn't equal zero so F wouldn't be tangent or normal to C at those points.

eseidl
 3 years ago
Best ResponseYou've already chosen the best response.1When x=y then F=n; the vectors are parallel when x=y.

math_proof
 3 years ago
Best ResponseYou've already chosen the best response.0thanks a lot for that explanation

eseidl
 3 years ago
Best ResponseYou've already chosen the best response.1no prob...been awhile since I've done these
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