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vectors.txt (1830B)

      1 ===============================================================================
      2 VECTORS
      3 ===============================================================================
      4 
      5 A vector is an object that has both magnitude (length) and direction and is
      6 often represented as an ordered list of numbers, like components along
      7 coordinate axes. In R^n, a vector is typically written as a column or row of n
      8 real numbers and can be added to other vectors or scaled by real numbers.
      9 
     10 
     11 -------------------------------------------------------------------------------
     12 1. BASIC VECTOR NOTATION (INLINE)
     13 -------------------------------------------------------------------------------
     14 
     15 A vector in 2D can be written as:
     16 
     17   v = (v1, v2)
     18 
     19 
     20 -------------------------------------------------------------------------------
     21 2. COLUMN VECTOR (DISPLAY)
     22 -------------------------------------------------------------------------------
     23 
     24 A column vector in 3D:
     25 
     26        | v1 |
     27   v =  | v2 |
     28        | v3 |
     29 
     30 
     31 -------------------------------------------------------------------------------
     32 3. VECTOR IN R^n
     33 -------------------------------------------------------------------------------
     34 
     35 In general, a vector in R^n is:
     36 
     37        | v1 |
     38        | v2 |
     39   v =  | .. |
     40        | vn |
     41 
     42 
     43 -------------------------------------------------------------------------------
     44 4. VECTOR ADDITION AND SCALAR MULTIPLICATION
     45 -------------------------------------------------------------------------------
     46 
     47 If u = (u1, u2) and v = (v1, v2), then:
     48 
     49   u + v = (u1 + v1, u2 + v2)
     50 
     51 and for a scalar a:
     52 
     53   a*v = (a*v1, a*v2)
     54 
     55 
     56 -------------------------------------------------------------------------------
     57 5. MAGNITUDE (LENGTH) OF A VECTOR
     58 -------------------------------------------------------------------------------
     59 
     60 The length of v = (v1, v2, v3) is:
     61 
     62   ||v|| = sqrt(v1^2 + v2^2 + v3^2)