Vector Analysis - Welcome to University of Malaya向量分析-欢迎来到马来亚大学.docVIP

Vector Analysis Summary of prerequisites (revision) A scalar quantity has magnitude only; a vector quantity has both magnitude and direction. The axes of reference, OX, OY, OZ, form a right-handed set. The symbols i, j, k denote unit vectors in the directions of OX, OY, OZ, respectively. If then The direction cosines are the cosines of the angles between the vector r and the axes OX, OY, OZ, respectively. For any vector r = axi+ayj+azk Scalar Product (Dot Product) where is the angle between A and B. If A = axi+ayj+azk and B = bxi+byj+bzk then Vector product (Cross Product) A x B = C =ABsinθn ; n is a unit vector in a direction perpendicular to A and B Angle between two vectors For perpendicular vectors For parallel vectors Solve!: If P is the point (3,2,6), determine r, , l, m, and n If A = 2i+3j+4k and B = i(2j+3k, find the direction cosines of each vector hence the angle between A and B If A = 2i(3j+4k and B = i+2j+5k, determine and verify If A = 3i(2j+4k and B = 2i(3j(5k, determine and verify Answers: a) b) c) d) Triple Product of three vectors If Then Remembering that We have Example: If Properties of scalar triple product a) Since interchanging two rows in a determinant reverses the sign. It can be shown i.e the scalar triple product is unchanged by a cyclic change of the vectors involved. b) i.e a change of vectors not in cyclic order, changes the sign of the scalar triple product c) since two rows are identical Coplanar vectors Vector triple products of three vectors : If Take note that It can be proved that: Example: If Equivalently using Try these: Determine for a) Answer: b) Answer: Differentiation of vectors Many practical problems deal with vectors that change with time (independent scalar) e.g. velocity, acceleration etc. If A can be represented as Differentiating with respect to t In general if u is the independent scalar parameter Example: A particle moves in space s