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Some theorems

Determinant notation. We will use the notation

|A B C|

to denote the determinant

whose columns are the components of the vectors

Theorem 1. If A, B, C are three linearly independent vectors in three-dimensional space and X is any other vector in 3-space then X can be expressed as a linear combination of vectors A, B and C i.e.

X = αA + βB + γC .

Furthermore, the vectors X, A, B, C are related by the following identity:

1) |A B C| X = |B C X| A + |C A X| B + |A B X| C

Proof. We pull from a magic hat the following set of three equations and assert them to be true.

Their validity follows from the fact for each equation, i = 1, 2, 3, the first row of the determinant is identical to one of the last three rows.

Let us now expand the determinant by the method of minors using the elements of the first row. We get

3) |B C X| a_{i} - |A C X| b_{i} + |A B X| c_{i} - |A B C| x_{i} = 0 i = 1, 2, 3

which is equivalent to

4) |A B C| X = |B C X| A + |C A X| B + |A B X| C

Thus since 2) is true and 4) is equivalent to 2), 4) is true.

End of Proof.

Corollary 1. If A, B, C are three linearly independent vectors in three-dimensional space and X is any other vector in 3-space then the vectors X, A, B, C are related by the following identity:

This follows from the fact that for a determinant |A B C|

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