Page:Linear Algebra (1882) Tevfik.djvu/22

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as if we knew $DE$ from its relation to $DM$. Therefore as $DM$ is admitted to be equal to $AM$ we have a right to assume that $AB$ equals $DE$.

Thus

$AB=DE$ and $AB+ED=0$.

And if

${\underline {AF}}=x.{\underline {DG}}$

we shall have

$AF=x.DG$.

12. It follows that, If a line $AB$ is represented by $a\alpha$ ($a$ being an abstract number, $\alpha$ a unit line in the direction $AB$), any line which is parallel to $AB$ or placed on the same line, and in the same direction and has the same length, can be designated also by $a\alpha$. In the case that the second line is in an opposite direction it will be designated by $-a\alpha$.

13. We have seen that $AB,BD,DH$ drawn successively in their respective directions, the line $AH$ which closes the polygon ABDH can be represented by

$AH=AB+BD+DH$,

or by designating the units of $AB,BD,DH$ respectively by $\alpha ,\beta ,\gamma$, and their lengths by $x,y,z$, and the line $AH$, by $\rho$, then

$\rho =x\alpha +y\beta +x\gamma$.

14. It is obvious that, if the lines $AB,BD,DH$ are not in the same plane we can consider the numbers $x,y$ and $z$ as cartisian coordinates of the point $H$.

When the directions $OX,OY,OZ$ are perpendicular one to the other, we shall use often $i,j,k$ to designate the linear units which are respectively in the directions $OX,OY,$ and $OZ$; if $x,y,z$ represent the rectangular coordinates of a point and $\rho$ the line which joins the origin $O$ to this point, we shall have

$\rho =xi+yj+zk$.

ADDITION.

13. If we take the lines $AB,AC,AD$ for example, and trace from the point $B$ a line equal to $AC$, and from the end of this a line equal to $AD$ and designate by AH the side which will close the polygon thus formed, the line $AH$ will be called the sum of the lines $AB,AG,AD$, or

$AH=AB+AC+AD$.

This operation we define as addition.

It will also be readily seen that the following operations

$AB+AD+AC$,

$AC+AB+AD$,

$AD+AC+AB$ etc.