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1. 3D Space and Vectors

1. 3D Space and Vectors - Math2011 3-Dimensional Space...

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Math2011, 3-Dimensional Space; Vectors 1.1 Rectangular Coordinates Points in the space are presented by their coordinates, similar to the case for points in a plane. Take three lines in the space which are mutually perpendicular and we call them the x-axis, y-axis and the z-axis. The intersection of these three lines is called the origin O . The xy-plane is the plane containing both the x-axis and the y-axis, and its similar for the yz-plane and the xz-plane. The first (x), second (y), third (z) coordinates of a point P are the perpen- dicular distances from P to the yz-plane, xz-plane, xy-plane respectively. For a function in one variable f , the graph of f is the set of points ( x, y ) in the plane satisfying y = f ( x ). The graph is a geometric object closely related to the function f . In general, if f is a function in two variables. The graph of f is the set of points ( x, y, z ) in the space satisfying z = f ( x, y ). It is a surface in the space and is closely related to the study of f . If ( a, b, c ) is a point in the space. The distance from the origin to this point is a 2 + b 2 + c 2 . This result is done by applying the Pythagoras theorem for twice. More generally, the distance between the points ( a, b, c ) and ( α, β, γ ) in the space is p ( a - α ) 2 + ( b - β ) 2 + ( c - γ ) 2 . Example 1.1 The set of points ( x, y, z ) in the space satisfying x 2 + y 2 + z 2 = 4 is the set of points having a distance 2 from the origin. Thus, it is a sphere with center the origin and radius 2 . Example 1.2 The set of points ( x, y, z ) in the space satisfying y 2 + z 2 = 4 is the set of points having a distance 2 from the x-axis. Thus, it is a cylinder with its axis the x-axis and radius 2 . 1.2 Vectors Definition 1.3 A vector is an ordered collection of numbers. There are some operations defined for vectors: Definition 1.4 Addition: ( x 1 , ..., x n ) + ( y 1 , ..., y n ) = ( x 1 + y 1 , ..., x n + y n ) 1
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Definition 1.5 Scalar multiplication: a ( x 1 , ..., x n ) = ( ax 1 , ...ax n ) Theorem 1.6 For all vectors x , y , z and for all numbers a , b , 1. ( a + b ) x = ax + bx 2. ( ab ) x = a ( bx ) 3. a ( x + y ) = ax + ay 4. x + y = y + x 5. ( x + y ) + z = x + ( y + z ) Definition 1.7 Inner product (Dot product): ( x 1 , ..., x n ) · ( y 1 , ..., y n ) = x 1 y 1 + ... + x n y n Theorem 1.8 For all vectors x , y , z , and for all numbers a , 1. ( ax ) · y = a ( x · y ) 2. ( x + y ) · z = x · z + y · z 3. x · y = y · x Vectors and their usual operations can be interpreted geometrically in the following way: 1. A vector ( x, y, z ) is represented by a point in the space whose coordinates are x , y and z . It can also be represented by an arrow pointing from the origin to the point with coordinates x , y , z . 2. The addition of two vectors u + v is the forth vertex of the parallelogram whose three other vertices are u , O and v . 3. The scalar multiplication by a number a is to lengthen vectors by a factor a . Definition 1.9 The length of a vector x is the non-negative number x · x . It is also denoted by || x || . Remark 1.10 The length of a vector || x || = x · x is the distance from the origin to the point representing x .
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