# difference of vectors

Let $\vec{a}$ and $\vec{b}$ be two vectors in the plane (or in a vector space).  The difference vector or difference$\vec{a}\!-\!\vec{b}$  of $\vec{a}$ and $\vec{b}$ is a vector $\vec{d}$  such that

 $\vec{b}+\vec{d}=\vec{a}.$

Thus we have

 $\displaystyle\vec{b}+(\vec{a}\!-\!\vec{b})=\vec{a}.$ (1)

According to the procedure of forming the sum of vectors by setting the addends one after the other, the equation (1) tallies with the picture below; when the minuend and the subtrahend emanate from a common initial point, their difference vector can be directed from the terminal point of the subtrahend to the terminal point of the minuend.

Remark.  It is easily seen that the difference $\vec{a}\!-\!\vec{b}$ is same as the sum vector

 $\vec{a}\!+\!(-\vec{b})$

where $-\vec{b}$ is the opposite vector of $\vec{b}$:  it may be represented by the directed line segment from the terminal point of $\vec{b}$ to the initial point of $\vec{b}$.

 Title difference of vectors Canonical name DifferenceOfVectors Date of creation 2013-03-22 17:47:19 Last modified on 2013-03-22 17:47:19 Owner pahio (2872) Last modified by pahio (2872) Numerical id 9 Author pahio (2872) Entry type Definition Classification msc 53A45 Synonym vector difference Synonym vector subtraction Related topic CommonPointOfTriangleMedians Related topic Difference2 Related topic ProvingThalesTheoremWithVectors Related topic VectorValuedFunction2 Defines difference vector Defines opposite vector