difference of vectors

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

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

Thus we have

$\overrightarrow{b}+(\overrightarrow{a}-\overrightarrow{b})=\overrightarrow{a}.$ (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 $\overrightarrow{a}-\overrightarrow{b}$ is same as the sum vector

$$\overrightarrow{a}+(-\overrightarrow{b})$$

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

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