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Introduction to Coordinates in Two Dimensions

We locate points in the plane by using two coordinate lines (instead of the single line that we used in one dimension). Refer to Fig. 1.6. We determine the coordinates of the given point P by first determining the x -displacement, or (signed) distance from the y -axis and then determining the y -displacement, or (signed) distance from the x -axis. We refer to this coordinate system as ( x , y )-coordinates or Cartesian coordinates . The idea is best understood by way of some examples.

Fig. 1.6

Example 1

Plot the points P = (3, −2), Q = (−4, 6), R = (2, 5), S = (−5, −3).

Solution 1

The first coordinate 3 of the point P tells us that the point is located 3 units to the right of the y -axis (because 3 is positive ). The second coordinate −2 of the point P tells us that the point is located 2 units below the x -axis (because −2 is negative). See Fig. 1.7 .

The first coordinate −4 of the point Q tells us that the point is located 4 units to the left of the y -axis (because −4 is negative ). The second coordinate 6 of the point Q tells us that the point is located 6 units above the x -axis (because 6 is positive). See Fig. 1.7 .

The first coordinate 2 of the point R tells us that the point is located 2 units to the right of the y -axis (because 2 is positive ). The second coordinate 5 of the point R tells us that the point is located 5 units above the x -axis (because 5 is positive). See Fig. 1.7 .

The first coordinate −5 of the point S tells us that the point is located 5 units to the left of the y -axis (because −5 is negative ). The second coordinate −3 of the point S tells us that the point is located 3 units below the x -axis (because −3 is negative). See Fig. 1.7 .

Fig. 1.7

Example 2

Give the coordinates of the points X, Y, Z, W exhibited in Fig. 1.8 .

Fig. 1.8

Solution 2

The point X is 1 unit to the right of the y -axis and 3 units below the x -axis. Therefore its coordinates are (1, −3).

The point Y is 2 units to the left of the y -axis and 1 unit above the x -axis. Therefore its coordinates are (−2, 1).

The point Z is 5 units to the right of the y -axis and 4 units above the x -axis. Therefore its coordinates are (5, 4).

The point W is 6 units to the left of the y -axis and 5 units below the x -axis. Therefore its coordinates are (−6, −5).

You Try It: Sketch the points (3, −5), (2, 4), ( , /3) on a set of axes. Sketch the set {( x , y ): x = 3} on another set of axes.

Example 3

Sketch the set of points ℓ = {( x , y ): y = 3}. Sketch the set of points k = {( x , y ): x = −4}.

Solution 3

The set ℓ consists of all points with y -coordinate equal to 3. This is the set of all points that lie 3 units above the x -axis. We exhibit ℓ in Fig. 1.9 . It is a horizontal line.

Fig. 1.9

The set k consists of all points with x -coordinate equal to −4. This is the set of all points that lie 4 units to the left of the y -axis. We exhibit k in Fig. 1.10 . It is a vertical line.

Fig. 1.10

Example 4

Sketch the set of points S = {( x , y ): x > 2} on a pair of coordinate axes.

Solution 4

Notice that the set S contains all points with x -coordinate greater than 2. These will be all points to the right of the vertical line x = 2. That set is exhibited in Fig. 1.11 .

Fig. 1.11

You Try It: Sketch the set {( x , y ): x + y < 4}.

You Try It: Identify the set (using set builder notation) that is shown in Fig. 1.12.

Fig. 1.12

Find practice problems and solutions for these concepts at Calculus Basics Practice Test.

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