Example 3.—The medians of a triangle tri sect each other. Take the origin at one vertex 0, and let the other vertices be a, b, then is and lb are mid-points, hence the equations of two niedians are r=a + x (lb—a), r=b y (ia—b). Making these equations simultaneous and equating coefficients of a and b, respectively (5), we get 1— x=ily,• ix=1—y, whence x=i y=1.
8. Rotors.—A rotor is a vector magnitude restricted to lie in some fixed straight line, called its line of position, or its axis. A rotor is completely specified by its magnitude, direc tion, sense, and one point on its axis; a familiar example is a force; it can be transferred to any point in its line of action, but cannot be moved out, of that line. Another example is a given angular velocity of rotation about a given axis; it can be specified by laying along the axis a segment whose length represents, on a conven ient scale, the magnitude of ,the angular velpc ity, and which points in the direction from which the rotation appears positive. This special kind of directed segment has given rise to the name rotor for any directed magni tude having an axis. The resultant of any number of rotors (of the same class) whose axes meet in a point P .is found by treating them as ordinary vectors, taking their vector sum and transferring it to the point P. A force R applied at P can be resolved into three components parallel to the three axes and acting at P, so -that R=iX j Y kZ.
9. Vector of an Area.—A plane area may be regarded as having magnitude and direction (orientation or facing), and can be represented by a vector normal to its plane, of length equal to the area on an assigned scale, and with an arrow pointing toward the side to which the area is supposed to face. If a given plane area is projected on any other plane, and if the pro jection is represented by a normal vector, this vector will be the projection of the vector of the given plane area.
10. Vector Sum for Faces of Polyhedron. — The sum of the vectors representing the faces of a closed polyhedron is zero, all of these being supposed to face outwards. For, if we project the entire surface on any plane, the part farthest from the plane projects into an area whose total vector may be represented by V, and the remainder of the surface projects into an area whose total vector is — V, pointing in the opposite direction. Thus the projection
of the original vector sum on the normal to the plane is zero; and since this is true for every plane, the original vector sum must itself be zero.
11. Scalar The scalar product of two vectors is the result obtained by multiplying the product of their lengths by the cosine of the angle between them. It may also be defined as the product of the length of either vector by the projection of the other upon it. The scalar product is denoted by writing the two vectors with a dot between them as a•b. This product evidently is commutative, so that a•b=b•a= ab cos 0, where a, b are the lengths (or tensors) of the vectors, and 0 the angle between their directions. If m and n are scalars, ma•nb= inn (a•13). If a and b are parallel, a•b=ab; and s•a=a'. If a and b are at right angles, s•b =0. The scalar multiplication table for the three fundamental unit vectors i, j, k is i•i=j-j=k-k=1.
i•j=j•i=j-k=k1=k•i=i•le =0 Scalar multiplication is distributive, i.e., (a+b)•c=a•c+b•c; this follows from the fact that the projection of a + b on the direction of c equals the sum of the projections of a and of b on the same direction.
12. Expansion of a• b.—If a=aii+asj-l-ahk, and b=bii+bsj-Fbsk, then, applying the distribu tive law and the multiplication table, we get a•b=a2b, a,b, + a,b,.
A trigonometrical interpretation is obtained by replacing a•13 by ab cos 0, dividing by ab, and noticing that as as bi cos — =cos pi, — w=cos —s= cos a ' a a bs bs b b= cos 13,, —= cos Ys where (al, Q,, y,) are the direction angles of a, and j9,, y,) are those of b. This gives the cosine formula, cos 0-- cos cos + cos cos ps + cos Yi cos As an application to two dimensions, we may take c=a + b as in the figure, then c•c= (a + b)• (a + b) =a-a + AO) + tra + b•b= a'+ 2 ab cos (r—c) -I- bi— 2 ab cos C, a fundamental relation in a triangle. For an application to mechanics let a, b, c rep resent forces applied at 0, and let r be their resultant, then r=a +,b +c. Let 0 be displaced to P, and let OP=d, then r•d=a• + 1•d + c•d; but led is the work done by the force a in the displacement d; hence the work done by a force in any displacement equals the sum of the amounts of work done by its components.