Page:Linear Algebra (1882) Tevfik.djvu/24

 And again if $$\alpha$$ and $$\beta$$ being in diifferent directions, we have

$a\alpha + b\beta = k\alpha + l\beta$

we must also have

$(a - k)\alpha + (b - l)\beta = 0$;

$$\therefore$$

$a - k = 0$ and $b - l = 0$.

19. If $$\alpha, \beta, \gamma$$ are non parallel lines in the same plane, it is always possible to find the numerical values of $$a, b, c,$$ so that,

$a\alpha + b\beta + c\gamma$ shall $= 0$.

For as these $$\alpha, \beta,$$ and $$\gamma$$ are on the same plane, a triangle can be constructed the sides of which shall be parallel respectively to $$\alpha, \beta, \gamma$$. Now if the sides of this triangle taken in order be

$a\alpha, b\beta, c\gamma$

we shall have, by going around the triangle,

$a\alpha + b\beta + c\gamma = 0$.

20. If $$\alpha, \beta, \gamma$$ are three lines neither parallel, nor in the same plane, it is impossible to find numerical values of $$a, b, c$$, not equad to zero, which shall render $$a\alpha + b\beta + c\gamma = 0$$, for $$a\alpha + b\beta$$ can be represented by a line in the plane parallel to $$\alpha, \beta$$. Now $$c\gamma$$ is not in that plane, therefore the sum of $$a\alpha + b\beta$$ and $$c\gamma$$ cannot equal $$0$$. It follows that, if $$a\alpha + b\beta + c\gamma = 0$$ and $$\alpha, \beta, \gamma$$ are not parallel to each other, they are in the same plane.

21 . There is but one way of making the sum of the multiples of $$\alpha, \beta, \gamma$$ equal to $$0$$.

Let

$a\alpha + b\beta + c\gamma = 0$

and also

$a,\alpha + b,\beta + c,\gamma = 0$.

By eliminating $$\gamma$$ we get

$(a c, - c a,)\alpha + (b c, - c b,)\beta = 0$;

but as $$\alpha, \beta$$ are in different directions,

$a c, - c a, = 0$ and $b c, - c b, = 0$;

$$\therefore$$

$a c, = c a$ and $b c, = c b,$

or

$a : b : c :: a, : b, : c,$,

so that the second equation is simply a multiple of the first. If we observe that the triangles which give the different values of $$a, b, c,$$ are similar the last proposition will be accepted a priori.

22. If $$\alpha, \beta, \gamma$$ are coinitial coplanar lines, terminating in a straight line, then the