Elements of Vector Analysis: Arranged for the Use of Students in Physics (Classic Reprint)

Excerpt from Elements of Vector Analysis: Arranged for the Use of Students in Physics(The three letters, i, j, k, will make an exception, to be mentioned more particularly hereafter.2. Def. - Vectors are said to be equal when they are the same both in direction and in magnitude. 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 A to B. We may use the notation 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 A and 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.Reversal of Direction, Scalar Multiplication and Division.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 un…