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

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—5—

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

Thus

and .

And if

we shall have

.

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

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

,

or by designating the units of respectively by , and their lengths by , and the line , by , then

.

14. It is obvious that, if the lines are not in the same plane we can consider the numbers and as cartisian coordinates of the point .

When the directions are perpendicular one to the other, we shall use often to designate the linear units which are respectively in the directions and ; if represent the rectangular coordinates of a point and the line which joins the origin to this point, we shall have

.

ADDITION.

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

.

This operation we define as addition.

It will also be readily seen that the following operations

,

,

etc.