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[parent] difference of vectors (Definition)

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

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


\begin{pspicture}(0,-1)(4,2.5) \psdot[linecolor=black](0,0) \psline[arrows=->,ar... ...[a](-1,0.7){$\vec{b}$} \rput[a](1.6,1.76){$\vec{a}\!-\!\vec{b}$} \end{pspicture}

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

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



"difference of vectors" is owned by pahio.
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See Also: common point of triangle medians, difference, proving Thales' theorem with vectors

Other names:  vector difference, vector subtraction
Also defines:  difference vector, opposite vector

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Cross-references: directed line segment, sum vector, terminal point, initial point, subtrahend, minuend, equation, sum of vectors, difference, vector space, plane, vectors
There are 4 references to this entry.

This is version 3 of difference of vectors, born on 2008-02-09, modified 2008-08-26.
Object id is 10249, canonical name is DifferenceOfVectors.
Accessed 979 times total.

Classification:
AMS MSC53A45 (Differential geometry :: Classical differential geometry :: Vector and tensor analysis)

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