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

 as if we knew $$D E$$ from its relation to $$D M$$. Therefore as $$D M$$ is admitted to be equal to $$A M$$ we have a right to assume that $$A B$$ equals $$D E$$.

Thus

$A B = D E$ and $A B + E D = 0$.

And if

$\underline{A F} = x. \underline{D G}$

we shall have

$A F = x. D G$.

12. It follows that, If a line $$A B$$ is represented by $$a\alpha$$ ($$a$$ being an abstract number, $$\alpha$$ a unit line in the direction $$A B$$), any line which is parallel to $$A B$$ 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 $$A B, B D, D H$$ drawn successively in their respective directions, the line $$A H$$ which closes the polygon ABDH can be represented by

$A H = A B + B D + D H$,

or by designating the units of $$A B, B D, D H$$ respectively by $$\alpha, \beta, \gamma$$, and their lengths by $$x, y, z$$, and the line $$A H$$, by $$\rho$$, then

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

14. It is obvious that, if the lines $$A B, B D, D H$$ 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 $$O X, O Y, O Z$$ are perpendicular one to the other, we shall use often $$i, j, k$$ to designate the linear units which are respectively in the directions $$O X, O Y,$$ and $$O Z$$; 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

$\rho = x i + y j + z k$.

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

$A H = A B + A C + A D$.

This operation we define as addition.

It will also be readily seen that the following operations

$A B + A D + A C$,

$A C + A B + A D$,

$A D + A C + A B$ etc.