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And by symmetry this point also lies on the other medians, and is a common point of trisection. (i) The joins of the mid points of opposite edges of a tetrahedron inter- sect and bisect each other. And by symmetry this centroid is also the point of bisection of the lines joining the mid points of the other pairs of opposite sides. 9, defining the position of the centroid, is to apply whether the associated numbers are positive or negative. (ii) The lines joining the vertices of a tetrahedron to the centroids of area of the opposite faces are concurrent. Consider the type of vector quantity whose magnitude is an area.
Hence the bisectors are concurrent at this centroid. But by symmetry this point must also lie on the bisectors of the angles at B and C. The centroid of the three points A, B, C with associated numbers a, b, c therefore lies on AP and divides it in the ratio (b c) : a. Finally, he would come to a very compact system in which vectors themselves and certain simple functions of vectors appeared, and would be delighted to find that the rules for the multiplication and general manipulation of these vectors were, considering the complexity of the Cartesian mathematics out of which he had discovered them, of an almost incredible simplicity. And the last part of this expression is zero because R divides AB in the ratio m : n. The forces on the particle due to the different centres ar« represented by — y - Ml . (ii) To find the vector equation of the plane through the point parallel to a and b. The vector CP is cop 1 ' • -^d (2) may therefore be written ' ' * ' -* a, is a variable number, positive LP=s& A ^ e opposite side of the origin as in the previous case. It is also sufficient ; for, assuming the condition satisfied in a linear relation between r, a, b and c, by making the coefficient of r unity, and denoting those of b and c by s and t, we obtain the relation in the form r=sb ic (l -s-t)&, showing that P is a point in the plane ABC. As an example of the use of this form of the vector equation of a plane consider the following : If any point within a tetrahedron ABOD is joined to the vertices, and AO, BO, CO, DO are produced to cut the opposite faces in P, Q, R, S respectively, then S-jp=l. The negative sign in (3) indicates that the torsion is regarded as positive when the rotation of the binormal as s increases is right-handed relative to the db has vector t. Let P be a moving point, and r its position vector relative to another point 0, which may be thought of as either moving or stationary. 183 184 VECTOR ANALYSIS The numbers refer to the articles.
But there would be no sign of a quaternion in his result, for one thing ; and, for another, there would be no metaphysics or abstruse reasoning required to establish i the rules of manipulation of his vectors." * This is the manner in which one would expect Vector Analysis to have originated. With as origin let the position vectors of A, B, C, D be a, b, c, d respectively. A glance at the figure shows that in this case , the opposite direction to n. If the tangent and the binormal at a point of a curve make angles 6, P. Then, since the sphere and curve have four points in common, the first three derivatives of c and R 2 with respect to s vanish. The velocity of P relative to is the rate of change of P's position relative to 0, and is therefore represented in measure and direction by the rate of change of r. where v=-j- is the speed of the particle along the path, and t is the unit tangent to the curve at that point. The vector representing the acceleration at the instant con- sidered is ^ v d dv dt ds dt ds dt or a, = ^-t KV 2 n, (2) at where k is the curvature and n the unit principal normal. But a glance at the figure shows that this vector is also OS = OP P# = a (b c) = (b c) a, and the argument obviously holds for any number of vectors. The commutative and associative laivs hold for the addition of any number of vectors. Thus the velocity of the wind is i - j, which is equivalent to 8 V% miles an hour from N. (2) If two concurrent forces are represented by n . OB respectively, their resultant is given by (m n) OR, ivhere R divides AB sothat n. Then g=tt is the rate of turning of OP, or , A A the angular velocity of P about 0.