School of Mathematics and Statistics, The University of Sydney
 7. Cartesian coordinates in three dimensions
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Length of a vector in terms of components

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The length of the vector -O-->Q in the XOY plane is  V~ --2---2-   x  + y.

  Y                   Q S  y  V~ x2-+-y2-    y   O        x              X                R

From the three dimensional picture below,

            Z             T                            P =  (x,y,z)             k                   j     z            O         y       S    Y           xi       R                Q   X

“cut out” the triangle OPQ to obtain:

                 P    ------------   V~  x2 + y2 + z2                  z        V~ -------- O      x2 + y2   Q

It can be seen that the length of the vector ---> OP is  V~ ------------   OQ2  +  QP 2 =  V~ ------------   x2 + y2 + z2.

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Main menu Section menu   Previous section Next section Cartesian form of a vector Addition and subtraction of vectors in terms of components