The University of Sydney - School of Mathematics and Statistics

Line vectors in two dimensions	Page 4 of 4

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.