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Is this text really indented as we would like it to be?
Hmm I need to check with netscape before deciding what's going on here!
This would be the very last line
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Problem : Find an equation for the line passing through the origin in the
direction of the vector (1, 2, 3) = i + 2j + 3k.
Let us first consider the following vectors : > with (mvcal) : demo(4.1) ; The line obtained from this formula can be seen with > demo(4.2) ; In general, if v is a vector then the line in the direction of v through the origin is given parametrically by L(t) = t v.
What about the equation of a line in the direction of (1, 2, 3) and through
the point (0, 1, 3) ? To understand what this formula means, let us first notice that (0,0,0),(1,2,3),2(1,2,3)=(2,4,6) and 3(1,2,3)=(3,6,9) are points on the original line. The points (0,1,3),(1,1,6),(2,3,9) and (3,5,12) are on the new line, they are obtained from:
etc.. This is how we get the "new" points from the "old" points. (To be more precise, when we do what we actually mean is . The end point of the position vector gives the coordinates of the "new" point on the "new" line. In general : If v is a vector and P is a point, then the line in the direction of v passing through P is given parametrically by: Here t is a parameter (a real number), so that tv is a scalar multiple of v. When we write +P we really mean + the vector whose initial point is the origin and whose terminal point is P. (See the example above.) The line is traced out by the endpoints of the vectors L(t) as t varies. L(t) is called a "vectorvalued function" because it is a rule which associates to each real number t a vector.
