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**Example text**

The equation for the plane wh ich passes through three H atoms in 24 ehapler 1 â€¢ Veclors a tetrahedral methane atom constitutes a good example. 24 ) through the point (1, 1, 1) in the plane. 25) The determinant format gives the components of the final vector but does not give the norm of the vector explicitly. This norm can be determined from the scalar products of the initial vectors. 29) Although the vector and scalar products are the more common vector products, a third product does exist. This is the direct product of two vectors.

Although the process becomes more difficult as the number of components increases, the method is still tractable for arbitrary N-dimensional spaces. Other methods can be used for vectors in two- and three-dimensional spaces. For example, if two orthogonal vectors are known in a three-dimensional space, the third vector is known using the vector product. 25 ) The problem can also be solved by using the two orthogonal vectors to genera te two equations in two unknowns. 26) The first component is assigned a value of 1 since the vector represents the relative magnitudes of the components.

12) is not orthogonal to 11) so it has a finite projection on 11). 4) The new vector 12') is the unit vector on the y axis since the x axis projection has been subtracted from the original vector 12). The technique also works if the vector 12) is selected as the initial vector. 23. The component of a linearly independent vector which is subtracted from this vector in a Gram-Schmidt orthogonalization. 9) Since vectors 11') and 12') are oriented +45 and - 45 with respect to the x axis, they are separated by 90 0 and are orthogonal.