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The rules for computing scalar products are such that only vectors with the same numbers of elements can be multiplied.

In analytic geometry, it can be shown that the scalar product of two vectors obeys the relationship:

b • c = (length of b) x (length of c) x cos 8 (2.9)

When the angle between two vectors is 8 = 90°, then cos 8 = 0 and the scalar product Orthogonal b • c = 0. As a consequence, two vectors whose scalar product is zero are orthogonal vectors (i.e. at right angle). This property will be used in Section 2.9 to compute eigenvectors.

A matrix whose (column) vectors are all at right angle of each other is called orthogonal.

Numerical example. Returning to the above example, it is possible to multiply each row of each monthly matrix with the correction vector (scalar product), in order to compare total monthly fish abundances. This operation, which is the product of a vector by a matrix, is a simple extension of the scalar product (eq. 2.8). The product of the July matrix B with the correction vector c is written as follows:

Numerical example. Returning to the above example, it is possible to multiply each row of each monthly matrix with the correction vector (scalar product), in order to compare total monthly fish abundances. This operation, which is the product of a vector by a matrix, is a simple extension of the scalar product (eq. 2.8). The product of the July matrix B with the correction vector c is written as follows: