notes

cross_product.txt (5789B)

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 ===============================================================================
 CROSS PRODUCT
 ===============================================================================
 
 The cross product is an operation that takes two vectors in R^3 and returns
 another vector in R^3, written a x b. Unlike the dot product, the result is a
 vector, not a scalar. The cross product is only defined in three dimensions
 (and in a generalized sense in seven dimensions; here we restrict to R^3).
 
 
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 1. GEOMETRIC MEANING
 -------------------------------------------------------------------------------
 
 Direction:
   a x b is perpendicular to both a and b, following the right-hand rule: if you
   point your fingers along a and curl them toward b, your thumb points in the
   direction of a x b.
 
 Magnitude:
   ||a x b|| = ||a|| * ||b|| * sin(theta)
 
   where theta is the angle between a and b. The length equals the area of the
   parallelogram spanned by a and b.
 
 
 -------------------------------------------------------------------------------
 2. ALGEBRAIC DEFINITION
 -------------------------------------------------------------------------------
 
 For vectors:
 
        | a1 |         | b1 |
   a =  | a2 |    b =  | b2 |
        | a3 |         | b3 |
 
 the cross product is:
 
           | a2*b3 - a3*b2 |
   a x b = | a3*b1 - a1*b3 |
           | a1*b2 - a2*b1 |
 
 This can be remembered using the determinant of a formal 3x3 matrix:
 
   | e1  e2  e3 |
   | a1  a2  a3 |  =  e1*(a2*b3 - a3*b2)
   | b1  b2  b3 |     - e2*(a1*b3 - a3*b1)
                      + e3*(a1*b2 - a2*b1)
 
 where e1, e2, e3 are the standard unit vectors in R^3.
 
 
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 3. THE "CROSS-OUT" METHOD (FASTEST)
 -------------------------------------------------------------------------------
 
 1. Stack them: write the components of the first vector over the second twice.
 
 2. Cross out the first and last columns.
 
 3. Multiply in an X pattern (top-left * bottom-right minus top-right *
    bottom-left) for each remaining pair:
 
     a1  a2  a3  a1  a2  a3
           \/  \/  \/
           /\  /\  /\
     b1  b2  b3  b1  b2  b3
 
     or:
 
     a1  b1 
     a2  b2
       \/
       /\
     a3  b3
       \/
       /\
     a1  b1
       \/
       /\
     a2  b2
     a3  b3
 
 Result:
 
           | a2*b3 - a3*b2 |
   a x b = | a3*b1 - a1*b3 |
           | a1*b2 - a2*b1 |
 
 
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 4. RULES OF CALCULATION (WITH EXAMPLES)
 -------------------------------------------------------------------------------
 
 Let a, b, c in R^3 and lambda in R.
 
 
 4.1 Anticommutativity
 
 Swapping the order flips the sign:
 
   a x b = -(b x a)
 
 Example:
 
   | 1 |   | 0 |       | 0 |
   | 0 | x | 1 |   =   | 0 |
   | 0 |   | 0 |       | 1 |
 
   | 0 |   | 1 |       |  0 |
   | 1 | x | 0 |   =   |  0 |
   | 0 |   | 0 |       | -1 |
 
 
 4.2 Distributivity over addition
 
   a x (b + c) = a x b + a x c
   (a + b) x c = a x c + b x c
 
 Example (second component of a x (b + c)):
 
        | 1 |       | 0 |       | 1 |
   a =  | 2 |  b =  | 1 |  c =  | 0 |
        | 0 |       | 1 |       | 1 |
 
   b + c = | 1 |
           | 1 |
           | 2 |
 
   a x b = |  2 |    a x c = |  2 |
           | -1 |            | -1 |
           |  1 |            | -2 |
 
   a x b + a x c = |  4 |
                   | -2 |
                   | -1 |
 
   a x (b + c) = | 2*2 - 0*1 |   |  4 |
                 | 0*1 - 1*2 | = | -2 |
                 | 1*1 - 2*1 |   | -1 |
 
 
 4.3 Scalar multiplication (homogeneity)
 
 A scalar can be factored out of either slot:
 
   (lambda*a) x b = a x (lambda*b) = lambda * (a x b)
 
 Example: a = | 1 |, b = | 0 |, lambda = 3
 
   (3*a) x b = | 3 |   | 0 |       | 0 |
               | 0 | x | 1 |   =   | 0 |
               | 0 |   | 0 |       | 3 |
 
             = 3 * | 0 | = 3 * (a x b)
                   | 0 |
                   | 1 |
 
 
 4.4 Cross product with the zero vector
 
   a x 0 = 0 x a = 0
 
 
 4.5 Parallel vectors
 
 a and b are parallel (or one is zero) if and only if:
 
   a x b = 0
 
 Example: a = | 2 |, b = | 1 | = (1/2)*a
 
              | 4 |       | 2 |
              | 6 |       | 3 |
 
   a x b = | 4*3 - 6*2 |   | 0 |
           | 6*1 - 2*3 | = | 0 |
           | 2*2 - 4*1 |   | 0 |
 
 
 4.6 Self-cross product
 
   a x a = 0
 
 (Special case of the parallel-vectors rule.)
 
 
 4.7 Jacobi identity
 
   a x (b x c) - b x (c x a) - c x (a x b) = 0
 
 
 4.8 Relation to dot product (vector triple product expansion)
 
   a x (b x c) = (a . c)*b - (a . b)*c
 
 Example: a = e1, b = e2, c = e3:
 
   a . c = 0,  a . b = 0
   =>  a x (b x c) = 0*b - 0*c = 0
 
   b x c = e1
   =>  e1 x e1 = 0
 
 
 4.9 Magnitude and angle
 
   ||a x b||^2 = ||a||^2 * ||b||^2 - (a . b)^2
 
 Equivalently:
 
   ||a x b|| = ||a|| * ||b|| * sin(theta)
 
 
 4.10 Relation to the dot product (scalar triple product)
 
   a . (b x c) = b . (c x a) = c . (a x b)
 
 This value is the (signed) volume of the parallelepiped spanned by a, b, c.
 
 Example:
 
   a = | 1 |  b = | 0 |  c = | 0 |
       | 0 |      | 1 |      | 0 |
       | 0 |      | 0 |      | 1 |
 
   b x c = | 1 |,   a . (b x c) = 1
           | 0 |
           | 0 |
 
 
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 5. WORKED EXAMPLE
 -------------------------------------------------------------------------------
 
 Compute u x v for:
 
        |  2 |         |  1 |
   u =  | -1 |    v =  |  4 |
        |  3 |         | -2 |
 
   u x v = | (-1)*(-2) - 3*4 |   |  2 - 12 |   | -10 |
           | 3*1 - 2*(-2)    | = |  3 + 4  | = |   7 |
           | 2*4 - (-1)*1    |   |  8 + 1  |   |   9 |
 
 Check:
 
   u . (u x v) = 2*(-10) + (-1)*7 + 3*9 = -20 - 7 + 27 = 0
   v . (u x v) = 1*(-10) + 4*7 + (-2)*9 = -10 + 28 - 18 = 0
 
 So the result is perpendicular to both u and v.