6. Scalar and vector triple product 6.1. The scalar triple product. Definition 1.37 (Scalar triple product). Let a; b; c 2 R3. The scalar triple product of the vectors a, b, and c (in that order) is the scalar (a × b) · c 2 R. Remark 1.38. We usually omit the brackets, since a × b · c can only mean (a × b) · c, because a × (b · c) is the cross product of a vector and a scalar, which is not defined. The scalar triple product satisfies the properties (i) provided we keep the vectors a, b, and c in the same order, we can exchange the cross product and the dot product, thus (1.13) a × b · c = a · b × c; (ii) the scalar product is unaltered provided we keep a, b, and c in the same cyclic order, namely (1.14) a × b · c = b × c · a = c × a · b: These properties follow by writing out the left and right hand sides, for instance for the first identity a × b · c = c1(a2b3 − a3b2) + c2(a3b1 − a1b3) + c3(a1b2 − a2b1) a · b × c = a1(b2c3 − b3c2) + a2(b3c1 − b1c3) + a3(b1c2 − b2c1) and it easy to check that these two quantities agree. For the second, we have a × b · c = c1(a2b3 − a3b2) + c2(a3b1 − a1b3) + c3(a1b2 − a2b1) b × c · a = a1(b2c3 − b3c2) + a2(b3c1 − b1c3) + a3(b1c2 − b2c1) and again we can check that these two quantities agree. 6.2. Geometric Interpretation of scalar triple product. The scalar triple product has a nice geometric interpretation. Consider three vectors u; v; w 2 R3. We can form a parallelepiped (a squashed rectangular box) by w taking these vectors as adjacent edges. The volume of a parallelepiped is given by the area v of the base, times the perpendicular height. If we take the base to be the parallelogram with adjacent 0 u edges u and v, this parallelogram has area = ju×vj. Now n = u × v is a vector perpendicular to the base, so the perpendicular height is given by the w length of the projection of w on n. u × v Thus the perpendicular height is given by v n jw · u × vj w · = : jnj ju × vj 0 u Combining these observations, we see that the volume is given by jw · u × vj ju × vj = jw · u × vj: ju × vj In other words, we have the volume of the parallelepiped with sides u, v, and w is given by jw · u × vj. 24 Remark 1.39. Had we chosen a different base, for example the parallelogram with adjacent edges u and w, we would have found the volume to be ju × w · vj . But from property (1.14), this is the same as jw · u × vj. Example 1.40. Find the volume of the parallelepiped with adjacent edges OA; OB; OC where a = (1; 2; −1); b = (3; 1; 0); c = (2; 2; 5). Solution: 6.3. Vector triple product. Given vectors a; b; c 2 R3, the vector products (a × b) × c and a × (b × c) are called vector triple prod- ucts. We have seen on an exercise that in general (a × b) × c 6= a × (b × c). So the order in which we compute the cross products matters. We now examine these vector triple products more closely. a × b Let ` be the plane through a and b. Then n = a × b is perpendicular to this plane and a n × c = (a × b) × c is perpendicular to n again, so lies back in the plane `. But any vector in ` can be 0 written as a linear combination of a and b. b Therefore we expect (a × b) × c = sa + tb, for some scalars s and t. Similarly, a × (b × c) lies in the plane containing b and c, and so we expect a × (b × c) = ub + vc, for some scalars u and v. In fact, these vector triple products, are given by (1.15) (a × b) × c = (a · c)b − (b · c)a and (1.16) a × (b × c) = (a · c)b − (a · b)c: The identities (1.15) and (1.16) follow from a direct computation. For instance, we have (a × b) × c = (a2b3 − a3b2; a3b1 − a1b3; a1b2 − a2b1) × (c1; c2; c3) = (a3b1 − a1b3)c3 − (a1b2 − a2b1)c2; (a1b2 − a2b1)c1 − (a2b3 − a3b2)c3; (a2b3 − a3b2)c2 − (a3b1 − a1b3)c1 = (a · c)b1 − (b · c)a1; (a · c)b2 − (b · c)a2; (a · c)b3 − (b · c)a3 = (a · c)b − (b · c)a: 25 The second identity (1.16) follows from a similar computation. Remark 1.41. The identities (1.15) and (1.16) are easy enough to learn if you remember that (i) the vectors that appear in the expressions on the right hand side are the two vectors that are in brackets on the left hand side (ii) the middle vector on the left has a plus sign on the right, the other a minus sign (iii) the scalar coefficient of each vector on the right is the dot product of the remaining two vectors. Example 1.42. Find (a × b) × c, where a = (0; 1; 0), b = (0; 0; 1), and c = (2; 2; 3). Solution: Example 1.43. Given jaj = 2 and a · b = −3, find (a × b) × a. Solution: Example 1.44. Simplify ((a × b) × a) × b. Solution: 26 Summary of Chapter 1. Vectors Section 1 - 3 Background material: dot product, projections, lines (parametric and coordinate), planes. Section 4 The cross product a × b = (a2b3 − a3b2; a3b1 − a1b3; a1b2 − a2b1) is ? to a and b. The standard properties contained in Lemma 1.23: given vectors u; v; w 2 R3 and a scalar a 2 R (1) u · (u × v) = 0 = v · (u × v), (2) u × v = −v × u, (3)( au) × v = u × (av) = a(u × v), (4)( u + v) × w = u × w + v × w and u × (v + w) = u × v + u × w, (5) u × u = v × v = 0. Behaviour of standard unit vectors : e1 × e2 = e3, e2 × e3 = e1, e3 × e1 = e2 [cyclic order]. Geometrical interpretation of the cross product: length: a × b = jajjbj sin θ; direction: perpendicular to a and b, given by right hand rule and a × b = 0 () a = tb for some t 2 R; () either a and b are parallel, or one of them is trivial. Section 5 1 Area of triangle ∆ABC = 2 j(b − a) × (c − a)j. Area of parallelogram ABCD = j(b − a) × (c − a)j. j(q−p)×vj Distance of a point Q from line p + tv is given by jvj . Plane through three points A, B, and C, has normal n = (b − a) × (c − a) and equation x · n = a · n. Two lines p + ta and q + sb are parallel () a × b = 0. j(q−p)·(a×b)j If a × b 6= 0, then the perpendicular distance between the lines p + ta and q + sb is ja×bj . Two lines p + ta and q + sb are called skew if a × b 6= 0, and they do not intersect. Section 6 Scalar triple product of a; b; c 2 R3 is the number (a × b) · c 2 R written a × b · c. The scalar triple product satisfies the properties (1.13) and (1.14): a × b · c = a · b × c and a × b · c = b × c · a = c × a · b: Volume of parallelepiped with sides a; b; c 2 R3 is given by ja × b · cj. Given a; b; c 2 R3 we define the vector triple products to be the vectors (a × b) × c and a × (b × c): (Note that (a × b) × c 6= a × (b × c)). The vector triple product satisfies the identities (1.15) and (1.16): (a × b) × c = (a · c)b − (b · c)a and a × (b × c) = (a · c)b − (a · b)c: 27 CHAPTER 2 MATRICES, LINEAR EQUATIONS AND DETERMINANTS 1. Matrices and matrix operations 1.1. Matrices. A matrix is a device for storing data. For example, if 4 teams A; B; C and D are playing in a rugby competition then the number of tries, penalties and conversions each team scores in their first game, could be recorded in the table below on the left. TPC 22 3 13 A 2 3 1 3 0 2 B 3 0 2 R = 6 7 64 1 37 C 4 1 3 4 5 0 3 0 D 0 3 0 However, once the format of the table is known, it is really only the array R of numbers on the right above that is important. The array R is called a matrix. Definition 2.1 (Matrices). A matrix is a rectangular array of numbers. The numbers in the array are called the entries in the matrix. Example 2.2. Some examples of matrices are 22 3 13 p 2 1 23 2 2 π e3 3 0 2 1 6 7 ; 3 0 ; 2 −1 0 ; 3 1 0 ; : 64 1 37 4 5 4 2 5 2 4 5 −1 2 0 0 0 0 3 0 The size of a matrix is given by the number of rows (horizontal lines) and columns (vertical lines) it contains. The first matrix in Example 2.2 has 4 rows and 3 columns, so its size is 4 by 3, denoted 4 × 3. The number of rows is always given first, followed by the number of columns. The other matrices in Example 2.2 are 3 × 2; 1 × 3; 3 × 3 and 2 × 1 matrices respectively. We use capital letters A, B, C, ::: for matrices, and small letters for their entries.