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

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

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

.

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 . If the lines are found equal in length, it is evident the length of will diminish with the angle ; and finally will become zero whenever this angle does; in this case the point coincides with , and the line with ; for this reason

or .

Thus in the expressions

or

and 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

or ,

hence if the line AB is represented by , the line will be .

10. If are on the same right line, and in the same direction, we admit, as in Numerical Algebra, that is to as to , that is

.

Now if , then and consequently

.

11. If are parallel in the same direction, and , we must admit

.

For if we take on the same right line , and , we admit (art. 10),

but compared to is situated exactly as compared to , and this similarity of position is so complete that if we know from its relation to it will be exactly