Summation subscript notation can be a very useful way of representing vectors and tensors, particularly for Cartesian coordinates.

2. Summation subscript notation can be a very useful way of representing vectors and tensors, particularly for Cartesian coordinates. In this notation we represent the components of a vector a for example as ai where the index i takes the values of 1, 2 or 3 to denoting the 3 components in the three coordinate directions. So, for example, we would write the vector sum c = a + b as ci=ai+bi. This notation can be used to write lengthy expressions in very compact form by means of the summation convention, which states that "if the same subscript appears twice, implicitly or explicitly, in a term of an expression, then the term is summed over that index". For example, if we have an expression that contains the term ...+aibjcj+... then this is interpreted as ...+ai(b1c1+b2c2+b3c3). An example of an 'implicit' repetition is bk2 = bkbk=|b|2. We can therefore write the dot-product of two vectors as a.b = aibi. The cross product also has a compact representation. Instead of c =axb we write ck= eijkaibj. Here, eijk is a commonly used device called the 'alternating tensor' which is defined as being equal to 1 if ijk are 123, 231 or 312, equal to -1 if ijk are 321, 213 or 132, and zero otherwise. (a) Show that the condition for three vectors, a, b and c, to be co-planar is eijkaibjck=0. (b) Show that the determinant of a 3 by 3 matrix Aij is given by eijkA1iA2jA3k