Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/36

It is generally in this way that the value of a vector is specified, viz., in terms of three known vectors. For such purposes of reference, a system of three mutually perpendicular vectors has certain evident advantages.

11. Normal systems of unit vectors.—The letters $i,j,k$ are appropriated to the designation of a normal system of unit vectors, ie., three unit vectors, each of which is at right angles to the other two and determined in direction by them in a perfectly definite manner. We shall always suppose that $k$ is on the side of the $i{\text{-}}j$ plane on which a rotation from $i$ to $j$ (through one right angle) appears counter-clockwise. In other words, the directions of $i,j$, and $k$ are to be so determined that if they be turned (remaining rigidly connected with each other) so that $i$ points to the east, and $j$ to the north, $k$ will point upward. When rectangular axes of ${\text{X}},{\text{Y}}$, and ${\text{Z}}$ are employed, their directions will be conformed to a similar condition, and $i,j,k$ (when the contrary is not stated) will be supposed parallel to these axes respectively. We may have occasion to use more than one such system of unit vectors, just as we may use more than one system of coordinate axes. In such cases, the different systems may be distinguished by accents or otherwise.

12. Numerical computation of a geometrical sum,—If

$\rho =a\alpha +b\beta +c\gamma ,$

$\sigma =a'\alpha +b'\beta +c'\gamma ,$

etc.,

then

$\rho +\sigma +{\text{etc.}}=(a+a'+{\text{etc.}})\alpha +(b+b'+{\text{etc.}})\beta +(c+c'+{\text{etc.}})\gamma ,$

i.e., the coefficients by which a geometrical sum is expressed in terms of three vectors are the sums of the coefficients by which the separate terms of the geometrical sum are expressed in terms of the same three vectors.

Direct and Skew Products of Vectors.

13. Def.—The direct product of $\alpha$ and $\beta$ (written $\alpha .\beta$) is the scalar quantity obtained by multiplying the product of their magnitudes by the cosine of the angle made by their directions.

14. Def—The skew product of $\alpha$ and $\beta$ (written $\alpha \times \beta$) is a vector function of $\alpha$ and $\beta$. Its magnitude is obtained by multiplying the product of the magnitudes of $\alpha$ and $\beta$ by the sine of the angle made by their directions. Its direction is at right angles to $\alpha$ and $\beta$, and on that side of the plane containing $\alpha$ and $\beta$ (supposed drawn from a common origin) on which a rotation from $\alpha$ to $\beta$ through an arc of less than 180° appears counter-clockwise.

The direction of $\alpha \times \beta$ may also be defined as that in which an ordinary screw advances as it turns so as to carry $\alpha$ toward $\beta$.