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# The Parametric Equation of a Line Determined by a Vector and a Point

 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 : 0.9(1,2,3), 2(1,2,3), 2.5(1,2,3), 3.2(1,2,3), -1.3(1,2,3), -2.2(1,2,3), etc.. (They are multiples of the vector (1, 2, 3).) The following Maple command will show the positions of their endpoints together with the vector (1, 2, 3) (without the arrow head) : > 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) ? Geometrically, we have to translate the previous line to the point (0, -1, 3). Mathematically, this means that the line becomes: t(1,2,3) + (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: (0, 0, 0) + (0,-1, 3) = (0, -1, 3), (1, 2, 3) + (0, -1, 3) = (1, 1, 6), (2, 4, 6) + (0, -1, 3) = (2, 3, 9) etc.. This is how we get the "new" points from the "old" points. (To be more precise, when we do (1, 2, 3) + (0, -1, 3) = (1, 1, 6), what we actually mean is (1i+2j+3k) + (0i-1j+3k) = (1i+1j+6k) . 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: L(t) = t v + P 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 "vector-valued function" because it is a rule which associates to each real number t a vector.

Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Mon Feb 12 15:01:36 EST 1996