Page:Linear Algebra (1882) Tevfik.djvu/21

 7. The operation of tracing the line $$A B$$ from the point $$A$$, and the line $$B D$$ from the point $$B$$, and giving to them their respective directions, will be represented by the expression $$A B + B D$$; and to show that by this operation $$A D$$ is found, the expression $$A B + B D = A D$$ will be employed.

8. If in departing from the point A the lines $$A B, B D, D H, H N, N O$$, are successively traced in their respective directions, the line joining $$A$$ to $$O$$, or $$A O$$, will be represented by

$A O = A B + B D + D H + N O$.

It is readily seen that after having traced AB, if in place of tracing the other lines in the order given, we trace successively a line parallel and equal in length to each one of these lines, in their respective directions, in whatever order, we shall still find the same line A O. It is needless to say that this manner of representation of straight lines is general.

9. It is now apparent what in Linear Algebra is meant by $$A B + B D = A D$$. If the lines $$A B, B D$$ are found equal in length, it is evident the length of $$A D$$ will diminish with the angle $$A B D$$; and finally $$A D$$ will become zero whenever this angle does; in this case the point $$D$$ coincides with $$A$$, and the line $$B D$$ with $$B A$$; for this reason

$A B + B D = 0$ or $AB + BA = 0$.

Thus in the expressions

$A B + B A$ or $B A + A B$

$$A B$$ and $$B A$$ neutralize each other; therefore when a line measured in one direction is represented by a positive symbol, the same line measured in the opposite direction may be represented by the same symbol taken negatively, that is

$A B = - B A$ or $B A = - A B$,

hence if the line AB is represented by $$\rho$$, the line $$B A$$ will be $$- \rho$$.

10. If $$A B, D E$$ are on the same right line, and in the same direction, we admit, as in Numerical Algebra, that $$A B$$ is to $$D E$$ as $$\underline{A B}$$ to $$\underline{D E}$$, that is

$A B = \frac{\underline{A B}}{\underline{D E}}D E$.|undefined

Now if $$\underline{A B} = \underline{D E}$$, then $$A B = D E$$ and consequently

$A B + E D = 0$.

11. If $$A B, D E$$ are parallel in the same direction, and $$\underline{A B} = \underline{D E}$$, we must admit

$A B = D E$.

For if we take $$A N, D M$$ on the same right line $$A D$$, and $$\underline{A N} = \underline{D M}$$, we admit $$A N = D M$$ (art. 10),

but $$D E$$ compared to $$D M$$ is situated exactly as $$A B$$ compared to $$A N$$, and this similarity of position is so complete that if we know $$A B$$ from its relation to $$A N$$ it will be exactly