Calculus- Early Transcendentals, 2021a

444 Three Dimensions
Exercise 12.5.19 Find the distance from (2,2,2) to x + y + z = −1.
Exercise 12.5.20 Find the distance from (2,−1,−1) to 2x − 3y + z = 2.
Exercise 12.5.21 Find the distance from (2,−1,1) to 〈2,2,0〉 +t〈1,2,3〉.
Exercise 12.5.22 Find the distance from (1,0,1) to 〈3,2,1〉 +t〈2,−1,−2〉.
Exercise 12.5.23 Find the cosine of the angle between the planes x + y + z = 2 and x + 2y + 3z = 8.
Exercise 12.5.24 Find the cosine of the angle between the planes x − y + 2z = 2 and 3x − 2y + z = 5.
12.6 Other Coordinate Systems
Coordinate systems are tools that let us use algebraic methods to understand geometry. While the rectangular
(also called Cartesian) coordinates that we have been discussing are the most common, some
problems are easier to analyze in alternate coordinate systems.
A coordinate system is a scheme that allows us to identify any point in the plane or in three-dimensional
space by a set of numbers. In rectangular coordinates these numbers are interpreted, roughly speaking, as
the lengths of the sides of a rectangular “box.”
In two dimensions you are already familiar (from Section 12.2) with an alternative, called polar coordinates.
In this system, each point in the plane is identified by a pair of numbers (r,θ). The number
θ measures the counter-clockwise angle between the positive x-axis and a vector with tail at the origin
and head at the point, as shown in Figure 12.15; the number r measures the distance from the origin to
the point. Either of these may be negative; a negative θ indicates the angle is measured clockwise from
the positive x-axis instead of counter-clockwise, and a negative r indicates the point at distance |r| in the
opposite of the direction given by θ.
The relationship between polar and rectangular coordinates is given by
and
x = r cosθ
y = r sinθ
r = √ x 2 + y 2
tanθ = y x
Example 12.15: Rectangular to Polar Coordinates
Convert the point (x,y)=(1, √ 3) into polar coordinates.
Solution. First calculate r:
r = √ x 2 + y 2 = √ 1 + 3 = 2