14. Line Vectors





If (_{1},v_{1}) and (_{2},v_{2}) are line vectors in two dimensions, then often (but not always) it is possible to combine them into a single equivalent line vector (,R), where R = v_{1} + v_{2}. Suppose that the lines _{1} and _{2} have a point P in common. That is P is on both _{1} and _{2}. This will always happen when _{1} and _{2} are not parallel and it is also true when _{1} and _{2} are the same line. Let R = v_{1} + v_{2} and let be the line through P in the direction of R. In this case the moments of (_{1},v_{1}) and (_{2},v_{2}) about P are both 0 and so is the moment of (,R) about P. Therefore the single line vector (,R) is equivalent to the system (_{1},v_{1}), (_{2},v_{2}).
Now suppose that _{1} and _{2} are distinct parallel lines and that R = v_{1} + v_{2} is not the zero vector. We can choose a point P_{1} on _{1} and a point P_{2} on _{2} such that the line m through P_{1} and P_{2} is perpendicular to both _{1} and _{2}.
Let be a unit vector in the direction of v_{1}. Then there are scalars v_{1} and v_{2} such that v_{1} = v_{1} and v_{2} = v_{2}. Put w = and suppose that = aw. Then = + = (a + 1)w and so av_{1} + (a + 1)v_{2} = 0. From this it follows that a = v_{2}/(v_{1} + v_{2}) and therefore, for any point O we have
Thus P divides the line segment P_{1}P_{2} in the ration v_{2} : v_{1}.
CouplesIf the vectors v_{1} and v_{2} have equal magnitude but opposite direction, then v_{1} + v_{2} = 0 and it is not possible to find a single line vector that is equivalent to the pair (_{1},v_{1}) and (_{2},v_{2}). In this case the system of two line vectors (_{1},v_{1}) and (_{2},v_{2}) is called a couple. When the forces applied to a rigid body form a couple, the effect is pure twist without any translation.


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