{"id":4870,"date":"2014-03-01T06:05:19","date_gmt":"2014-02-28T21:05:19","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4870"},"modified":"2014-08-01T18:54:57","modified_gmt":"2014-08-01T09:54:57","slug":"vector-algebra","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4870","title":{"rendered":"Vector algebra"},"content":{"rendered":"
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The operations of addition, subtraction and multiplication familiar in the algebra of numbers are, with suitable definition, capable of extension to an algebra of vectors. The following definitions are fundamental. <\/p>\n

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  1. Two vectors \\bold{A} and \\bold{B} are equal<\/em> if they have the same magnitude and direction regardless of their initial points. <\/li>\n
  2. A vector having direction opposite to that of vector \\bold{A} but with the same magnitude is denoted by -\\bold{A}. <\/li>\n
  3. The sum<\/em> or resultant<\/em> of vectors \\bold{A} and \\bold{B} is a vector \\bold{C} formed by placing the initial point of \\bold{B} on the terminal point of \\bold{A} and joining the initial point of \\bold{A} to the terminal point of \\bold{B}. The sum \\bold{C} is written \\bold{C} = \\bold{A} + \\bold{B}. The definition here is equivalent to the parallelogram law<\/em> for vector addition. <\/li>\n
  4. The difference<\/em> of vectors \\bold{A} and \\bold{B}, represented by \\bold{A} - \\bold{B}, is that vector \\bold{C} which added to \\bold{B} gives \\bold{A}. Equivalently, \\bold{A} - \\bold{B} may be defined as \\bold{A} + (-\\bold{B}). If \\bold{A} = \\bold{B}, then \\bold{A} - \\bold{B} is defined as the null<\/em> or zero vector<\/em> and is represented by the symbol \\bold{0}. This has a magnitude of zero but its direction is not defined. <\/li>\n
  5. Multiplication of vector \\bold{A} by a scalar m<\/em> produces a vector m\\bold{A} with magnitude |m| times the magnitude of \\bold{A} and direction the same as or opposite to that of \\bold{A} according as m is positive or negative. If m = 0, m\\bold{A} = \\bold{0}, the null vector. <\/li>\n<\/ol>\n<\/blockquote>\n