School of Mathematics and Statistics, The University of Sydney
 14. Line Vectors
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Line vectors in two dimensions

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If (l1,v1) and (l2,v2) are line vectors in two dimensions, then often (but not always) it is possible to combine them into a single equivalent line vector (l,R), where R = v1 + v2.

Suppose that the lines l1 and l2 have a point P in common. That is P is on both l1 and l2. This will always happen when l1 and l2 are not parallel and it is also true when l1 and l2 are the same line.

Let R = v1 + v2 and let l be the line through P in the direction of R. In this case the moments of (l1,v1) and (l2,v2) about P are both 0 and so is the moment of (l,R) about P. Therefore the single line vector (l,R) is equivalent to the system (l1,v1), (l2,v2).

                     l1          v1                            l                     R  P           v            2          l2

Now suppose that l1 and l2 are distinct parallel lines and that R = v1 + v2 is not the zero vector. We can choose a point P1 on l1 and a point P2 on l2 such that the line m through P1 and P2 is perpendicular to both l1 and l2.

   m P  2                         l2                                 l P1                         1

Let v be a unit vector in the direction of v1. Then there are scalars v1 and v2 such that v1 = v1v and v2 = v2v. Put w = -P--->P  1 2 and suppose that -P--->P    1 = aw.

Then ----> P P2 = ----> P P1 + ----> P1P2 = (a + 1)w and so av1 + (a + 1)v2 = 0. From this it follows that a = -v2/(v1 + v2) and therefore, for any point O we have

---> OP = ----> OP1 + ----> P1P
= -O-P-->    1 - aw
= ----> OP1 + --v2--- v1 + v2(----> OP2 -----> OP1)
=   ---->      ----> v1OP1--+-v2OP2-     v1 + v2.

Thus P divides the line segment P1P2 in the ration v2 : v1.

Couples

If the vectors v1 and v2 have equal magnitude but opposite direction, then v1 + v2 = 0 and it is not possible to find a single line vector that is equivalent to the pair (l1,v1) and (l2,v2). In this case the system of two line vectors (l1,v1) and (l2,v2) is called a couple. When the forces applied to a rigid body form a couple, the effect is pure twist without any translation.

      v          - v

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