7. Cartesian coordinates in three dimensions





Given two vectors in Cartesian form the sum + = is obtained by completing the parallelogram. It can be proved that this is the same as the following calculation:
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).
SubtractionThe rule for subtraction works in exactly the same way. Writing for the vector , the rule above gives
The subtraction is illustrated below. Recall that =  = .
An example of the use of this rule is the calculation of the Cartesian form of the position vector of a point P_{2} relative to a point P_{1}. Suppose that in Cartesian form, = x_{1}i + y_{1}j + z_{1}k and = x_{2}i + y_{2}j + z_{2}k. Then


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