Multivariate Calculus - Bruce E. Shapiro

Lecture 1
Cartesian Coordinates
We express a point in three dimensions as an ordered sequence (x, y, z). In Cartesian
coordinates, the third axis (the z-axis) is raised perpendicular to the x-y plane
according to the right-hand rule:
1. Construct the x-y plane on a piece of paper.
2. Place your right hand on top of the paper at the origin with your fingers
curling from the x-axis toward the y-axis.
3. Your thumb should be pointing upwards out of the paper. This is the direction
of the z-axis.
Figure 1.1: Cartesian coordinates.
Definition 1.1 (Distance Formula) The distance between two points P 1 = (x 1 , y 1 , z 1 )
and P 2 = (x 2 , y 2 , z 2 ) is calculated using the following distance formula
|PQ| = √ (x 1 − x 2 ) 2 + (y 1 − y 2 ) 2 + (z 1 − z 2 ) 2 (1.1)
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