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.

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

The vector is xi. The vector is yj. The vector is the sum of and , that is,

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.

Draw a perpendicular PT from P to the OZ axis.

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

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

The formula

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

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