School of Mathematics and Statistics, The University of Sydney
 7. Cartesian coordinates in three dimensions
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Addition and subtraction of vectors in terms of components

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Given two vectors in Cartesian form

-O-->P  = ai + bj + ck  -O-->Q  = di + ej + f k,

the sum - --> OP + ---> OQ = - --> OR is obtained by completing the parallelogram.

           Z                  P              R                            Q           O                    Y    X

It can be proved that this is the same as the following calculation:

--->     --->    ---> OR  =  OP  + OQ     =  (ai + bj + ck) + (di + ej + fk)      =  (a + d)i + (b + e)j + (c + f)k

That is, the components of a sum are the sums of the components.

Notice that the parallelogram OQRP is part of a two dimensional plane sitting within three dimensional space (in a tilted way like the slanting face of the roof of a house).

Subtraction

The rule for subtraction works in exactly the same way. Writing -O-->S for the vector ---> OP ----> OQ, the rule above gives

-O-->S  =  -O-->P  - -O-->Q      =  (ai + bj + ck) -  (di + ej + fk)     =  (a - d)i + (b- e)j + (c-  f)k

The subtraction is illustrated below. Recall that ---> OS = ---> OP ----> OQ = - --> QP.

            Z                   P  S                            Q                                Y             O    X

An example of the use of this rule is the calculation of the Cartesian form of the position vector -P--->P  1 2 of a point P2 relative to a point P1. Suppose that in Cartesian form, ----> OP1 = x1i + y1j + z1k and ----> OP2 = x2i + y2j + z2k.

Then

-- -->    ---->    ----> P1P2  = P1O  + OP2           ---->    ---->       = - OP1 +  OP2        = - x1i- y1j - z1k + x2i + y2j + z2k       = (x2 - x1)i + (y2 - y1)j + (z2 - z1)k

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