By Harrison D.M.

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Because we have given signs to the sides A and B, the vector area comes endowed with a direction. There are physical applications where it is convenient to be able to give a direction to an area [see Eq. 11)]. / t Parallelogram FIG. 18 (a) Vector area of parallelogram = AB IsinOI C IS C =A X B AXB1~~ Volume of a parallelepiped. The scalar I(A X B) 0 CI / = V / / is the volume of the parallelepiped of which A X B is the area of the base and C the slant height or edge (Fig. 18b). If the three vectors A, B, and C lie in the same plane, the volume will be zero; thus three vectors are coplanar if and only if (A X B) C = o.

But we are faced with a problem: Our position is -fixed by that of the earth and we are not yet free to wander through space with instruments to measure astronomical triangles. How can we test the validity of euclidean geometry to describe measurements in space? 7 Gauss measured the angles of a tnangle with vertices on three mountain tops and found no deviation from 180 0 within the accuracy of his measurements. Planetary Predictions A first lower limit of about 5 X 10 17 cm to the radius of curvature of our own universe is implied by the consistency of astronomical observations within the solar system.

A vector does not necessarily have location, although a vector may refer to a quantity defined at a particular point. Two vectors can be compared even though they measure a physical quantity FIG. 3 The vector 06r of magnitude 0 6r IS In the direction of r and IS p o / FIG. 4 The vector of r. Note that r = rr. r IS the u nit vector In the direction , 30 Chapter 2 A -" FIG. 5 .. (a) Vector A t B (b) Vector B (c) The vector sum A + B. (d) The vector sum B + A IS equal to A +B Vectors defined at different points of space and time.