18 VECTOK ANALYSIS. 



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 a = /3. 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. (Some- 

 times 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 a may be used to denote the magnitude of the 

 vector a. 



Addition and Subtraction of Vectors. 



6. Def. The sum of the vectors a, /3, etc. (written a+/3+etc.) is 

 the vector found by the following process. Assuming any point A, 

 we determine successively the points B, C, etc., so that AB = a, 

 BC = /3, etc. The vector drawn from A to the last point thus deter- 

 mined 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 a, /3, etc. are called the components. 



