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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. He might introduce tentatively an abbreviated nota- tion for these combinations. P n are n points dividing the circumference of a circle into n equal parts. , AP n ivhere A is any point, not necessarily in the plane of the circle. (4) A particle is acted on by a number of centres of force, some of which attract and some repel, the force in each case varying as the distance, W. Show that the resultant passes through a fixed point for all positions of the particle. From the symmetry of this result the second theorem follows. If P is any point on the plane its position vector OP is coplanai EQUATION OF PLANE •27 $§19,20] with a and b, and may therefore be resolved into components parallel to these and expressed in the form r = sa i! Any point on the plane is given by (1) for some values of s and t ; and for all values of these variables the point sa tb lies on the given plane. at j ons (i^ (2) and (3) involve each two variable numbers - 1 ° analogous to the equations of the lines considered 'Hs form of the equation for a plane is not, however, These two lines iiior is it the most convenient. r the scalar product of two vectors has been second theorem follows. sb tc - r = 0, If P is any point on the plane iu algebraic sum of the coefficients I§ 20, 21] GEOMETRICAL APPLICATIONS 29 of the vectors is zero. The necessary and sufficient condition that a linear relation, connecting the position vectors of any number of fixed points, should be independent of the origin, is that the algebraic sum of the coefficients is zero. It is a well-known geometrical result that the area A' of the orthogonal projection of a plane figure of area A, upon another plane inclined at an angle 6 to it, is A cos 6. The centre of curvature C then lies on the principal normal, so that PC = P n=-n. as The first term is zero because n is perpendicular to b, and the db equation then shows that -r- is perpendicular to t. This rate of turning of the binormal is called the torsion of the curve at the point P. After a little practice he would perceive the laws according to which these combinations arose and how they operated. Find the resultant of forces represented — by AP lt AP 2 , . Let P be the position of the particle, and O r , 0.,, ■ ■ ■ those of the centres of force. We may therefore speak of (1) as the vector equation of the given plane. connecting them (1) Let c be the position vector of C, and r tb' the given plane. This is the necessary and sufficient condition that four points should be coplanar,* That it is necessary has just been proved. In virtue of the relation (1) we may write the equation, of OP as u r = y (mb wc pi) (3) This line intersects the plane BCD at a point for which (2) and (3) give identical values for r ; and since b, c, d are non-coplanar vectors, this requires From these, by addition, we have u = l/(m n p), showing 'that the ratio Qp u l AP 1 u l m n p The other ratios may be written down by cyclic permutation of the symbols, and their sum is obviously equal to unity. If then a, a' are the vectors representing the vector areas A, A' respectively, a' is the resolute of a in the direction normal to the plane of projection ; for a' =a cos 6. It is, •of course, the arc-rate of rotation of the plane of curvature, since b is perpendicular to this plane.
It is apparent then that the processes of current vector analysis have sprung from the work of Hamilton and Grassmann. If the resultant of two forces is equal in magnitude to one of the components, and perpendicular to it in direction, find the other component. Two forces act at the corner A of a quadrilateral ABCD, —*■ — ■ represented by AB and AD ; and two at C represented by OB and CD. of particles of masses 1, 2, 3, 4, 5, 6, 7, 8 grams respectively, placed at the corners of a unit cube, the first four at the corners A, B, C, D of one face, and the last four at their projec- tions A', B', C, D' respectively on the opposite face. And by symmetry this is also a point of quadrisection of the lines joining the other vertices to the centroids of area of the opposite faces. But as the direction might be either of two opposite directions along the normal, some convention is necessary. The line of intersection is perpendicular to both n and n', so that , n and perpendicular to the plane rc = q ? Show that the plane containing the two straight lines r - a = ia' and r - a ' = sa is represented by [raa'] = 0. I-b 1-c m'b nrc n*b n'c IV.] EXERCISES ON CHAPTER IV. The shortest distances between a diagonal of a rectangular parallelepiped, whose sides are a, b, c, and the edges not meeting it, are be ca ab Vtf c 2 ' Vc 2 a*' Va? The shortest distance between two opposite edges of a regular tetrahedron is equal to half the diagonal of the square described on an edge. Find the shortest distance between the straight lines r = (k and r - a = sb, and determine the equation of the line which cuts both at right angles. Show that the perpendicular distance of the point a from the line whose Pliicker's coordinates are d, m is mod (m d cos \fs cos =r s r ^ (2) t d , . and dt * (&y\ 2 f^Y K _ W/ \ds») \ds 2 ) \d8*)' If R is any point in the plane of curvature, the vectors R-r, t and n are coplanar.
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.
The sum is independent of the order and the grouping of the terms. But if P', Q' are points in OP and OQ respectively so that OP' : OP = OQ': OQ=m, then P'Q' is parallel to PQ and m times it in length. Three or more vectors are said to be coplanar when a plane can be drawn parallel to all of them ; otherwise they are non- coplanar. Let r, s be unit vectors §§ 67, 68] KINEMATICS 101 parallel and perpendicular to r, the latter in the positive direction A A of &). (1) A particle P moves in a plane with constant angular velocity w about 0. Instantaneous axis, 87 ; velocity, 55, 65 ; acceleration, 55, 66.
He could do so by careful inspection and comparison of the Cartesian formulae. • -)PG, where 67 is the centroid of the points O v 2 , 3 , . And this is a fixed point indepen- dent of the position of P. Also calculate the module and direction cosines of each. If the position vectors of P and Q are i 3 j - 7k and 5i - 2j 4k y respectively, find PQ and determine its direction cosines. If the vertices of a triangle are the points a 1 i a 2 ] a 3 k, & 1 i H-& 2 j & 3 k and c-,i c 2 j c 3 k, — ■* what are the vectors determined by its sides ? \ -■ — y Express AC, DB, BC and CA in terms of a and b. If a, b are the vectors determined by two adjacent sides of a regular hexagon, what are the vectors determined by the other sides taken in order ? A point describes a circle uniformly in the i, j plane taking 12 seconds to complete one revolution. The velocity of a boat relative to the water is represented by 3i 4j, and that of the water relative to the earth by i-3j. Find the velocity of the first relative to the second when the radius to the latter makes an angle 6 with the direction of motion of the former, 6 increasing. s = l-t; l-s = t; l-s = l-t, which are satisfied by s = t = \. Again the centroid of area of ABC is the point J (a b c) ; and the line joining D to this point has the equation r = s(a b c). And no point off the plane can be represented by (2). so that the plane is one through A parallel to b - a and c - a. , a„ be the position vectors of the fixed points A t , A 2 , . The normal vector PP' bears to this direction of rotation the same relation as the trans- lation to the direction of rotation p of a right-handed screw. With this convention a vector area may be represented by a vector normal to the plane of the figure, in the direction relative to which it is positive, and with module equal to the measure of the area. Hence if r is resolved into two components in the plane of a and r, one parallel and the other perpendicular to a, these components are a-r a and r a*r respectively. Its radius is called the radius of curvature, and its centre the centre of curvature. The straight line through P perpen- dicular to the plane of curvature is called the binormal.
He would find certain combinations of symbols and quantities occurring again and again, usually in systems of threes. AG, where G is the centroid of the points P v P 2 , ■ ■ ■ , P n - And by symmetry G coincides with the centre of the circle. e 18 VECTOR ANALYSIS [ CH - and the intensities for different centres being different. If its initial position vector relative to the centre is i, and the rotation is from i to j, find the position vectors at the end of 1, 3, 5, 7 seconds ; also at the end of 1J and 4| seconds. In the previous exercise find the velocity vectors of the moving point at the end of 1|, 3 and 7 seconds. What is the velocity of the boat relative to the earth if i and j represent velocities of one mile an hour E. [Assume that variable velocities obey the same law of composition as uniform velocities. Also the centroid of area of J) AC is the point (a c) ; and the line joining this to £ is r = ib ( l_ ? These two lines intersect at the point for which s = t = \ that is the point J(a b c). (1) To find the vector equation of the plane through the origin parallel to a and b. ■where s, t are numbers which vary as the point P moves over the plane. To find the equation of the plane through the three points A, B, C whose position vectors are a, b, c, we observe that ^4. "*3S equation is therefore lme j r = a s(b-a) £(c-a) r = (l-s-f)a sb *c (3) Also the ot. 23 are coplanar ; and if the (1) To find the vector equg their position vectors is written parallel to a and b. The sum of two vector areas represented by a and b is defined to be the vector area represented by a b. Similarly, if any vector r is resolved into components parallel to the unit vectors i, j, k, these components are i'ri, j-rj, k*rk respectively, and r is their sum. This circle clearly lies in the osculating plane at P ; and we leave it as an exercise for the student to show that the radius of curvature p is given by p = -. It is parallel to the vector b = t*n, and t, n, b form a right-handed system of mutually perpendicular unit vectors. Since b is a vector of constant length, it follows, as in Art. Further, by differentiating the § GO] CURVATURE AND TORSION OF A CURVE 87 relation t'b = 0, we find «:n'b 1* t- = 0. perpendicular to b, and must therefore be parallel to n. .(3) And just as in formula (2) the scalar k measures the arc-rate of turning of the unit vector t, so here X measures the arc-rate of turning of the unit vector b.
In advanced three-dimensional work in nearly every branch of . Gibbs's work on dyadics is the most original and important part of his theory, and will be found of great service in our second volume. Since the second derivative 88 VECTOR ANALYSIS [CH. Then, in virtue of (2), we find ds ~dr dh dh~ _ds ds 2 ds 3 _ d 3 r dr ds 3 ds so that R = dr d 2 r d 3 i ds 2 ds 3 . It may also be put in the form k 2 X ds 1 1 d_fl\ h k A ds\K/ ■(5) This gives the well-known formula R2 ~K 2 X 2 \ds k) ■ .(6) 23. Find the radius of spherical curvature for the curve in Exer- cise 14. A variable point P on a given curve has a position vector r relative to a fixed origin 0. The former is the rate of increase of the speed and is independent of the shape of the curve. If 66 is the angle turned through by the tangent at P during the interval dt, -j- is the rate of rotation of the tangent.