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Extra info for Séminaire Pierre Lelong Année 1970-1971
Then IlxAIl = Ixl IIAII (absolute value of x times the norm of A). Proof By definition, we have IIxAII 2 = (xA)·(xA), which is equal to by the properties of the scalar product. Taking the square root now yields what we want. Let S 1 be the sphere of radius 1, centered at the origin. Let a be a number > 0. If X is a point of the sphere S 1, then aX is a point of the sphere of radius a, because II aX II = allXIl = a. In this manner, we get all points of the sphere of radius a. ) Thus the sphere of radius a is obtained by stretching the sphere of radius 1, through multiplication by a.
Thus de/dt = wand e = wt + a constant. For simplicity, assume that the constant is O. Then we can write the position of the bug as X(e) = X(wt) = (cos wt, sin wt). If the angular speed is 1, then we have simply the representation X(t) = (cos t, sin t). Example 2. If the bug moves around a circle of radius 2 with angular speed equal to 1, then its position at time t is given by X(t) = (2 cos t, 2 sin t). More generally, if the bug moves around a circle of radius r, then the position is given by X(t) = (r cos t, r sin t).
Let P = (cla,O) and A = (-bla, 1). We see that an arbitrary point (x, y) satisfying the equation ax + by = c can be expressed parametrically, namely (x,y) = P + tAo In higher dimensions, starting with a parametric representation x = P + tA, we cannot eliminate t, and thus the parametric representation is the only one available to describe a straight line. 36 [1, §6] VECTORS I, §5. EXERCISES 1. Find a parametric representation for the line passing through the following pairs of points. (a) P 1 = (1, 3, -1) and P 2 = (-4, 1,2) (b) P 1 =(-1,5,3) and P 2 =(-2,4,7) Find a parametric representation for the line passing through the following points.