PlanetMath: difference of vectors

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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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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 MSC:

53A45 (Differential geometry :: Classical differential geometry :: Vector and tensor analysis)

Pending Errata and Addenda

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