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

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We begin with two dimensions. We have the following picture illustrating how to construct the Cartesian form of a point Q in the XOY plane.

Y S Q = (x,y) y j x X O i R

Vectors i and j are vectors of length 1 in the directions OX and OY respectively.

The vector -O-->R is xi. The vector -O-->S is yj. The vector -OQ--> is the sum of -OR--> and -O-->S, that is,

---> OQ = xi + yj.

We now extend this to three dimensions to show how to construct the Cartesian form of a point P. Define k to be a vector of length 1 in the direction of OZ. We now have the following picture.

Z P k O Y j i X

Draw a perpendicular PT from P to the OZ axis.

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

In the rectangle OQPT,PQ and OT both have length z. The vector -O-->T is zk. We know that - --> OQ = xi + yj. The vector - --> OP, being the sum of the vectors ---> OQ and ---> OT, is therefore

-O-->P = -O-->Q + -O-->T = xi + yj + zk.

This formula, which expresses ---> OP in terms of i, j, k, x, y and z, is called the Cartesian representation of the vector ---> OP in three dimensions. We call x, y and z the components of ---> OP along the OX, OY and OZ axes respectively.

The formula

- --> OP = xi + yj + zk.

applies in all octants, as x, y and z run through all possible real values.

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