doesn't the normal plane defines all the tangents or directional derivatives at the point t of a 3 dimensional curve?
How come the normal vector of the normal plane is the derivative of the curve r(t)?
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calculus
1 Answers
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I'm not sure what you mean about directional derivatives; a curve is a function of just one variable, so we can't really take directional derivatives, and there is only one tangent, which is the derivative.
Think about it: a normal plane has to be perpendicular to the curve. What vector could we use as the normal vector for the plane? It should be tangent to the curve, because if the vector is tangent to the curve and the plane is normal to the vector, then the plane will be normal to the curve. Well, the derivative works as a tangent vector.
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0err curve r(t) you're right it's not 3d, but how come r'(t) is the normal vector of the plane? – 2012-11-28
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0@Math: Well, that's what my second paragraph says. $r'(t)$ is tangent to the curve. If we want our plane to be normal to the curve, it should be normal to a vector that is tangent to the curve. Try to picture it in your head or maybe make a drawing. That usually helps. – 2012-11-28
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0what does normal mean? – 2012-11-28
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0http://www.google.ca/imgres?imgurl=http://www.learner.org/courses/mathilluminated/images/units/8/1832.png&imgrefurl=http://www.learner.org/courses/mathilluminated/units/8/textbook/06.php&h=205&w=254&sz=3&tbnid=n90AH7mks6uUzM:&tbnh=90&tbnw=112&zoom=1&usg=__3KFZApdc-UZVLfAudq8RXdRu9HE=&docid=0T6fKkyHfxl8iM&sa=X&ei=EoO1UNTjKsfr0QHfsYHIDQ&ved=0CFQQ9QEwBQ&dur=1325 – 2012-11-28
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0it's kinda confusing because it's not "normal" as you say, yet they say it is normal. – 2012-11-28
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0@Math: in geometry, normal usually means perpendicular. The drawing you show is confusing; the plane looks like it contains the normal vector, while it should be perpendicular to it (and therefore tangent to the surface). – 2012-11-28
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0the drawing is wrong right? – 2012-11-28
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0Actually, I think I read it wrong. The plane is parallel to the vector and therefore normal to the surface. But here we have a surface and a vector that's normal to it, while your question was about curves. – 2012-11-28