I. Vectors and Geometry in Two and Three （即在两个和三个向量和几何）.pdf

I. Vectors and Geometry in Two and Three Dimensions §I.1 Points and Vectors Each point in two dimensions may be labeled by two coordinates (a, b) which specify the position of the point in some units with respect to some axes as in the figure on the left below. Similarly, each point in three dimensions may be labeled by three coordinates (a, b, c). The set of all points in two dimensions is z y (a, b, c) (a, b) c b a y x b a x denoted IR2 and the set of all points is three dimensions is denoted IR3 . The distance from the point (x, y, z) ′ ′ ′ ′ 2 ′ 2 ′ 2 to the point (x , y , z ) is (x − x ) + (y − y ) + (z − z ) so that the equation of the sphere centered on (1, 2, 3) with radius 4 is (x − 1)2 + (y − 2)2 + (z − 3)2 = 16. A vector is a quantity which has both a direction and a magnitude, like a velocity or a force. To specify a vector in three dimensions you have to give three components, just as for a point. To draw the vector with components a, b, c you can draw an arrow from the point (0, 0, 0) to the point (a, b, c). Similarly, to z y (a, b, c)