Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/34

18 2. Def.—Vectors are said to be equal when they are the same both in direction and in magnitude. This equality is denoted by the ordinary sign, as $$\alpha = \beta$$. The reader will observe that this vector equation is the equivalent of three scalar equations. A vector is said to be equal to zero, when its magnitude is zero. Such vectors may be set equal to one another, irrespectively of any considerations relating to direction.

3. Perhaps the most simple example of a vector is afforded by a directed straight line, as the line drawn from $$\text{A}$$ to $$\text{B}$$. We may use the notation $$\overline{\text{AB}}$$ to denote this line as a vector, i.e., to denote its length and direction without regard to its position in other respects. The points $$\text{A}$$ and $$\text{B}$$ may be distinguished as the origin and the terminus of the vector. Since any magnitude may be represented by a length, any vector may be represented by a directed line; and it will often be convenient to use language relating to vectors, which refers to them as thus represented.

4. The negative sign ($$-$$) reverses the direction of a vector. (Sometimes the sign $$+$$ may be used to call attention to the fact that the vector has not the negative sign.)

Def.—A vector is said to be multiplied or divided by a scalar when its magnitude is multiplied or divided by the numerical value of the scalar and its direction is either unchanged or reversed according as the scalar is positive or negative. These operations are represented by the same methods as multiplication and division in algebra, and are to be regarded as substantially identical with them. The terms scalar multiplication and scalar division are used to denote multiplication and division by scalars, whether the quantity multiplied or divided is a scalar or a vector.

5. Def.—A unit vector is a vector of which the magnitude is unity.

Any vector may be regarded as the product of a positive scalar (the magnitude of the vector) and a unit vector.

The notation $$\alpha_{0}$$ may be used to denote the magnitude of the vector $$\alpha$$.

6. Def.—The sum of the vectors $$\alpha, \beta$$, etc. (written $$\alpha + \beta + \text{etc.}$$) is the vector found by the following process. Assuming any point $$\text{A}$$, we determine successively the points $$\text{B, C}$$, etc., so that $$\overline{\text{AB}} = \alpha, \overline{\text{BC}} = \beta$$, etc. The vector drawn from $$\text{A}$$ to the last point thus determined is the sum required. This is sometimes called the geometrical sum, to distinguish it from an algebraic sum or an arithmetical sum. It is also called the resultant, and $$\alpha, \beta$$, etc. are called the components.