14. Line Vectors
 Glossary Engineering Examples

### Line vectors in two dimensions

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If (1,v1) and (2,v2) 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 = v1 + v2.

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 = v1 + v2 and let be the line through P in the direction of R. In this case the moments of (1,v1) and (2,v2) 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,v1), (2,v2).

Now suppose that 1 and 2 are distinct parallel lines and that R = v1 + v2 is not the zero vector. We can choose a point P1 on 1 and a point P2 on 2 such that the line m through P1 and P2 is perpendicular to both 1 and 2.

Let be a unit vector in the direction of v1. Then there are scalars v1 and v2 such that v1 = v1 and v2 = v2. Put w = and suppose that = aw.

Then = + = (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

 = + = - aw = + ( -) = .

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 (1,v1) and (2,v2). In this case the system of two line vectors (1,v1) and (2,v2) 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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