13.1-2 Geometry & Vectors

13.1-2 Geometry & Vectors - Vector algebra....

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ET 12.1; M 13.1. Geometry Distance in R 3 . For Cartesian coordinates in R 2 , one application of Pythagoras’s theorem shows that the (shortest) distance between two points is p x ) 2 + (Δ y ) 2 . In R 3 , two applications of that theorem show that the distance is p x ) 2 + (Δ y ) 2 + (Δ z ) 2 . Just as the equation ( x - a ) 2 + ( y - b ) 2 = r 2 ( ) gives a circle in R 2 , the corresponding equation ( x - a ) 2 + ( y - b ) 2 + ( z - c ) 2 = r 2 gives a sphere in R 3 . But in R 3 , equation ( ) gives a cylinder parallel to the z -axis. Similarly, the equation y = 2 x - 3 gives a line in R 2 , but it gives a plane in R 3 . 1
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ET 12.2; M 13.2. Vectors Change in position. Displacement vectors de- scribe changes in position. These vectors are de- termined by the differences between the coordinates of the final and initial positions. Those differences are the (standard) components of the vector. The change from the origin to a point is the position vector of that point. The components or that position vector are the coordinates of the point.
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Unformatted text preview: Vector algebra. Combining two changes of posi-tion corresponds to adding components. Vectors can also be subtracted, and there is a ~ 0 vector. Multi-plying vectors by numbers other than 1 rescales the vectors and sometimes reverses them. Vector algebra has many of the properties of algebra with numbers. Magnitude and direction. The length of a vector is the square-root of the sum of the squares of its (standard) components. Each vector, except ~ 0, can be factored in the form ~ b = b b , where the number b is the magnitude of the vector ~ b , and the unit vector b gives the direction of ~ b . Standard basis vectors. All vectors in R 3 are linear combinations of the unit vectors i , j and k . The End 8 January 2007...
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13.1-2 Geometry & Vectors - Vector algebra....

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