7. Cartesian coordinates in three dimensions
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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 P2 relative to a point P1. Suppose that in Cartesian form, = x1i + y1j + z1k and = x2i + y2j + z2k. Then
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