By Gron O., Nass A.
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Additional info for Einstein's theory - A rigorous introduction to general relativity for the mathematically untrained
Hence |B|·|A | = |A|·|B | or B · A = A · B. Properties 3 and 4 follow immediatley from Eq. 8) and Fig. 12. Properties 5 and 6 are definitions. Property 7 is shown by considering Fig. 14. 14: Associative rule seen that |(A + B) | = |A | + |B |. Multiplying each term by the number |D| gives |D|·|(A+ B) | = |D|·|A |+|D|·|B |. Hence D · (A + B) = D · A + D · B. Due to the property 2 it also follows that (A + B) · D = A · D + B · D. 9) We can now express the dot product of A and B by the components of A and B.
18: Parallelogram extended by A and B In Fig. 18 we have drawn the parallelogram defined by the vectors A and B. We have also drawn a dotted line from the end-point of vector B and perpendicular to the vector A. If the triangle OQB is moved to the other side of the parallelogram, a rectangle QPRB is formed with the same area as the triangle. The area of the rectangle, and thus of the parallelogram, is the length of A times the height QB. This height is just the component of B perpendicular to A.
4 Calculus of vectors. 4 Calculus of vectors. 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.