Vector Field
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The position vector
Let P, Q be points with coordinates [Equation goes here - download the original pdf to see it.] and [Equation goes here - download the original pdf to see it.] referred to axes Oxyz. The position vector of Q relative to P is the vector [Equation goes here - download the original pdf to see it.] To prove that this is a vector we have to show (a) that it is a triad of scalars, (b) remains invariant under the translation of axes, and (c) obeys the transformation law. Proof (a) Clearly [Equation goes here - download the original pdf to see it.] is a triad of scalars if [Equation goes here - download the original pdf to see it.] and [Equation goes here - download the original pdf to see it.] are. (b) Let the origin be moved to a position [Equation goes here - download the original pdf to see it.] relative to O. Then the points P and Q relative to this new origin will be [Equation goes here - download the original pdf to see it.] and [Equation goes here - download the original pdf to see it.] hence [Equation goes here - download the original pdf to see it.] c) Let [Equation goes here - download the original pdf to see it.] be the transformation matrix of a rotation of axes from Oxyz to [symbol]. Then [Equation goes here - download the original pdf to see it.] Note, using the summation convention where [Equation goes here - download the original pdf to see it.] the last part of this proof becomes [Equation goes here - download the original pdf to see it.] illustrating the usefulness of the summation convention.
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Contents of
Vector Field
1 Scalar and Vector Algebra -Prerequisites
2 Formal definition of a vector
3 The position vector
4 About vectors
5 Orthonormal basis
6 Scalar product
7 Scalar invariant
8 Resolute of a vector
9 The cross (vector) product
10 Distributive law
11 Basis vectors
12 Area of the parallelogram
13 Parallel, anti-parallel
14 The triple scalar product
15 The triple vector product
16 Quadruple vector products
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