Schwere, Elektricität und Magnetismus:383

VECTOR ANALYSIS.

<section begin=t1 /> Vorlage:Idt2With this understanding we may say:—

Vorlage:Idt2The coefficients by which any vector is expressed in terms of three other vectors are the direct products of that vector with the reciprocals of the three.

Vorlage:Idt2Among other relations which are satisfied by reciprocal systems of vectors are the following:

Vorlage:MathForm2

Vorlage:MathForm2

(See No. 34.)

Vorlage:MathForm2

(See No. 29.)

Vorlage:Idt2A system of three mutually perpendicular unit vectors is reciprocal to itself, and only such a system.

Vorlage:Idt2The identical equation

Vorlage:MathForm2

may be regarded as a particular case of equation (2).

Vorlage:Idt2The system reciprocal to $\alpha \times \beta ,\beta \times \gamma ,\gamma \times \alpha$ is

Vorlage:MathForm2

Vorlage:Idt239. Scalar equations of the first degree with respect to an unknown vector.—It is easily shown that any scalar equation of the first degree with respect to an unknown vector $\rho$, in which all the other quantities are known, may be reduced to the form

Vorlage:MathForm2

in which $\alpha$ and $a$ are known. (See No. 35.) Three such equations will afford the value of $\rho$ (by equation (8) of No. 37, or equation (3) of No. 38), which may be used to eliminate $\rho$ from any other equation either scalar or vector.

Vorlage:Idt2When we have four scalar equations of the first degree with respect to $\rho$, the elimination may be performed most symmetrically by substituting the values of $\rho \cdot \alpha$ etc., in the equation,

Vorlage:MathForm1

which is obtained from equation (8) of No. 37 by multiplying directly by $\delta$. It may also be obtained from equation (5) of No. 37 by writing $\delta$ for $\rho$, and then multiplying directly by $\rho$.

Vorlage:Idt240. Solution of a vector equation of the first degree with respect to the unknown vector.—It is now easy to solve an equation of the form

Vorlage:MathForm2

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