Cross product | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-05-07

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Cross_product


00:04:15 1 Definition
00:07:13 2 Names
00:08:32 3 Computing the cross product
00:08:42 3.1 Coordinate notation
00:18:55 3.2 Matrix notation
00:21:19 4 Properties
00:26:09 4.1 Geometric meaning
00:26:18 4.2 Algebraic properties
00:29:13 4.3 Differentiation
00:29:28 4.4 Triple product expansion
00:29:43 4.5 Alternative formulation
00:30:01 4.6 Lagrange's identity
00:33:22 4.7 Infinitesimal generators of rotations
00:34:12 5 Alternative ways to compute the cross product
00:38:35 5.1 Conversion to matrix multiplication
00:39:45 5.2 Index notation for tensors
00:44:28 5.3 Mnemonic
00:47:25 5.4 Cross visualization
00:51:20 6 Applications
00:51:33 6.1 Computational geometry
00:53:05 6.2 Angular momentum and torque
00:53:16 6.3 Rigid body
01:00:38 6.4 Lorentz force
01:04:30 6.5 Other
01:08:05 7 Cross product as an external product
01:13:40 8 Cross product and handedness
01:14:02 9 Generalizations
01:18:27 9.1 Lie algebra
01:20:52 9.2 Quaternions
01:22:32 9.3 Octonions
01:23:49 9.4 Exterior product
01:24:17 9.5 External product
01:26:03 9.6 Multilinear algebra
01:27:28 9.7 Skew-symmetric matrix
01:27:47 10 History
01:28:29 11 See also
01:28:44 12 Notes
01:30:03 13 References
01:31:26 14 External links
01:31:58 Exterior product
01:32:52 External product
01:33:30 Multilinear algebra
01:43:46 Skew-symmetric matrix
01:47:42 History
01:51:27 See also



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SUMMARY
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In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space 
  
    
      
        
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    {\displaystyle \left(\mathbb {R} ^{3}\right)}
   and is denoted by the symbol 
  
    
      
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    {\displaystyle \times }
  . Given two linearly independent vectors 
  
    
      
        
          a
        
      
    
    {\displaystyle \mathbf {a} }
   and 
  
    
      
        
          b
        
      
    
    {\displaystyle \mathbf {b} }
  , the cross product, 
  
    
      
        
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    {\displaystyle \mathbf {a} \times \mathbf {b} }
  , is a vector that is perpendicular to both 
  
    
      
        
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    {\displaystyle \mathbf {a} }
   and 
  
    
      
        
          b
        
      
    
    {\displaystyle \mathbf {b} }
   and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product).
If two vectors have the same direction (or have the exact opposite direction from one another, i.e. are not linearly independent) or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The cross product is anticommutative (i.e., 
  
    
      
        
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          a
        
      
    
    {\displaystyle \mathbf {a} \times \mathbf {b} =-\mathbf {b} \times \mathbf {a} }
  ) and is distributive over addition (i.e., 
  
    
      
        
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        ×
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          b
        
        +
        
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        =
        
          a
        
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          b
  ...