This is an example in book by Howard Anton:(adsbygoogle = window.adsbygoogle || []).push({});

Vector form of line is ##\vec r=\vec r_0+t\vec v## where ##\vec v## is parallel with the line. So both ##\vec r## and ##\vec r_0## are POSITION VECTORS.

To change parameters,

1)Let u=t ##\Rightarrow\; \vec r=\vec r_0+u\vec v##.

2) ##\frac {d\vec r}{du}=\vec v\;\Rightarrow\;|\frac {d\vec r}{du}|=|\vec v|##

##s=\int_0^t |\frac {d\vec r}{du}|du=t|\vec v|\;\Rightarrow\; t=\frac{s}{|\vec v|}##

3)##\Rightarrow\; \vec r=\vec r_0+\frac{s}{|\vec v|}\vec v##

My question is in #2 above. In order for ##\frac {d\vec r}{du}=\vec v## which is the tangent vector of the curve traced by ##\vec r ##, ##\vec r ## has to be a VECTOR VALUE FUNCTION, NOT JUST A POSITION VECTOR. This means ##\vec r =\vec r(w)## where w is the independent variable that make the tip of ##\vec r## tracing out the line when w increases or decreases.( of cause it can be a vector value function of many variables also).

As you see, my problem is there are TWO parameters, t and w. The book only change parameter of t, which has nothing to do with the vector value function ##\vec r(w)##. t only tell the line is multiple of ##\vec v##. In another word, this example totally miss the point in changing parameter. The parameter needed to be change is w, not t.

Please comment on this.

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