By Gron O., Nass A.
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Extra resources for Einstein's theory - A rigorous introduction to general relativity for the mathematically untrained
If k = 0, we write kA = 0 and call 0 the null vector. The magnitude of the null vector is zero. If two vectors A and B are parallel, there exists a scalar k so that B = kA. Thus A = (1/k)B. One might also be tempted to put B/A = k. But division by a vector is not defined within the calculus of vectors. We now have the necessary tools for defining the component vectors of A. The definition may be stated as follows: the component vectors of A are the products of the components of A and the basis vectors.
They are said to be orthogonal to each other. ) We have a Cartesian coordinate system. The basis vectors along the x-axis and y-axis, are written ex and ey , respectively. The magnitude of a vector A is designated by |A| . The basis vectors ex and ey have per definition magnitude equal to 1, |ex | = |ey | = 1. 4 Calculus of vectors. 4 Calculus of vectors. Two dimensions 35 The quantities Ax and Ay in Fig. 6 will be called the components of the vector A. (Note that in the calculus of vectors it is usual to use the upper right suffix to select a component rather than to indicate the exponent of a power.
Two dimensions 42 in ch. ) The dot product of A and B is denoted by A · B and defined as the magnitude of A times the magnitude of B’s projection onto A. This is illustrated in Fig. 12. 12: Vector projection The magnitude of B’s projection onto A is denoted by |B |. ) Some properties of this product should be noted. 1. The product is a scalar quantity. 2. B · A = A · B. 4 Calculus of vectors. Two dimensions 43 3. For vectors of given magnitude, the product has a maximal value, equal to the magnitude of A times the magnitude of B, if the vectors have the same direction.
Einstein's theory - A rigorous introduction to general relativity for the mathematically untrained by Gron O., Nass A.