School of Mathematics and Statistics, The University of Sydney
 7. Cartesian coordinates in three dimensions
Main menu Section menu   Previous section Next section Coordinate axes and Cartesian coordinates Length of a vector in terms of components
Glossary
Assumed knowledge
Related topics
Examples

Cartesian form of a vector

Page 2 of 5 

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.

Feedback
Main menu Section menu   Previous section Next section Coordinate axes and Cartesian coordinates Length of a vector in terms of components