Scalar Multiplication of Vectors

For a real scalar $a$

and vector $\mathbf{u}$ , the product $a\mathbf{u}$ has magnitude

$\displaystyle \lVert a\mathbf{u}\rVert=\vert a\vert\,\lVert\mathbf{u}\rVert,$

and points in the same direction as $\mathbf{u}$ if $a>0$

and the opposite direction if $a<0$

The algebraic laws are

$\displaystyle (ab)\mathbf{u}=a(b\mathbf{u}),$

$\displaystyle (a+b)\mathbf{u}=a\mathbf{u}+b\mathbf{u},$

$\displaystyle a(\mathbf{u}+\mathbf{v})=a\mathbf{u}+a\mathbf{v}.$

For $a\ne0$ ,

$\displaystyle \frac{\mathbf{u}}{a}=\left(\frac1a\right)\mathbf{u}.$

Two nonzero vectors are parallel exactly when one is a scalar multiple of the other:

$\displaystyle \mathbf{u}=k\mathbf{v}.$

In modern linear-algebra language, vector addition and scalar multiplication make the set of free vectors into a vector space.

Modern notation references

The notation and terminology in this modernized article follow standard present-day mechanics and vector-analysis usage, particularly:

J. R. Taylor, classical mechanics, University Science Books, 2005.
D. Kleppner and R. Kolenkow, An Introduction to Mechanics, 2nd ed., Cambridge University Press, 2014.
H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison–Wesley, 2002.

Source

This article is a modernized restatement of the corresponding Public domain article in Louis Brand, Vectorial Mechanics, John Wiley & Sons, New York, 1930, Chapter I, “Vector Algebra.” The original 1930 edition is the source basis.

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