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Handbook of mathematics for engineers and scienteists part 23

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Straight lines dividing the interior angles of a triangle into three equal parts are called angle trisectors. 'The three points of intersection of adjacent angle trisectors of a triangle form an equilateral triangle (tig. 3.9). | 122 Analytic Geometry 3. Vectors a and b are collinear if and only if a X b 0. In particular a X a 0 and a a X b b a X b 0. 4. Aa X b a X Ab A a X b associativity with respect to a scalar factor . 5. The cross product of basis vectors is i x i j X j k X k 0 i X j k j X k i k X i j. 6. If the vectors are given by their coordinates a ax ay az and b bx by bz then i j k a X b ax ay az ay bz - az by i az bx - ax bz j ax by - ay bx k. 4.5.3.6 bx by bz 7. The area of the parallelogram spanned by vectors a and b is equal to s ia X bi r 1 ay az 1 1ax az 1 1ax ay 1 S a X b by bZ bX bz bx by 4.5.3.7 8. The area of the triangle spanned by vectors a and b is equal to S a X b 1. ay az I2 I ax az I2 I ax ay I2 S 2 a X b 2 I by bz I I bx bz I I bx by 4.5.3.8 Example 1. The moment with respect to the point O of a force F applied at a point M is the cross product of the position vector OM by the force F i.e. M OM X F. 4.5.3-3. Conditions for vectors to be parallel or perpendicular. A vector a is collinear to a vector b if b Aa or a X b 0. 4.5.3.9 A vector a is perpendicular to a vector b if a b 0. 4.5.3.10 Remark. In general the condition a b 0 implies that the vectors a and b are perpendicular or one of them is the zero vector. The zero vector can be viewed to be perpendicular to any other vector. 4.5.3-4. Triple cross product. The triple cross product of vectors a b and c is defined as the vector d a X b X c . 4.5.3.11 The triple cross product is coplanar to the vectors b and c it can be expressed via b and c as follows a X b X c b a c - c a b . 4.5.3.12 4.5. Coordinates Vectors Curves and Surfaces in Space 123 4.5.3-5. Scalar triple product of three vectors. The scalar triple product of vectors a b and c is defined as the scalar product of a by the cross product of b and c abc a b X c . 4.5.3.13 Remark. The scalar triple product of three vectors a b and c is also denoted by abc. Properties of scalar triple product 1. abc bca cab - bac - cba - acb . 2. a b cd acd bcd .