Section 2

Projections on the Coordinate Axes

The projection of a point on an axis is the place where a perpendicular dropped from the point meets that axis — like the shadow of the point cast straight down (or straight across). Projecting a whole line segment simply means projecting both of its endpoints.

Definition

Projection of a Point

On the x-axis

(x , y) → (x , 0)

Keep the x-coordinate, replace the y-coordinate with 0.

On the y-axis

(x , y) → (0 , y)

Keep the y-coordinate, replace the x-coordinate with 0.

xy-6-6-4-4-2-2224466A(3,4)A'(3,0)

Perpendicular to the x-axis gives A'(3 , 0).

xy-6-6-4-4-2-2224466A(3,4)A''(0,4)

Perpendicular to the y-axis gives A''(0 , 4).

Notice that

  • If a point lies on an axis, its projection on that axis is the point itself, and its projection on the other axis is the origin.
  • A(3 , 0) lies on the x-axis: its projection on the x-axis is A(3 , 0), and on the y-axis it is O(0 , 0).
  • B(0 , -8) lies on the y-axis: its projection on the y-axis is B(0 , -8), and on the x-axis it is O(0 , 0).

Example

Worked Example: Projecting Four Points

Example 1

Plot A(-2 , 3), B(4 , -1), C(0 , -3), D(2 , 0), then find the projection of each point on both axes.

xy-6-6-4-4-2-2224466AA'A''BB'B''

A(-2 , 3) → x-axis: A'(-2 , 0) · y-axis: A''(0 , 3)

B(4 , -1) → x-axis: B'(4 , 0) · y-axis: B''(0 , -1)

C(0 , -3) → x-axis: O(0 , 0) · y-axis: C(0 , -3) itself

D(2 , 0) → x-axis: D(2 , 0) itself · y-axis: O(0 , 0)

Definition

Projection of a Line Segment

The projection of a line segment on an axis is the projection of every one of its points on that axis. In practice: if A' is the projection of A and B' the projection of B, then A'B' is the projection of AB.

Length of the projection on the x-axis = | x₂ − x₁ | · Length on the y-axis = | y₂ − y₁ |

Rule 1

The projection length is less than or equal to the length of the segment itself.

Rule 2

If the segment is parallel to an axis, its projection on that axis equals its own length.

Rule 3

If the segment is perpendicular to an axis, its projection on that axis is a single point (length 0).

Examples

Worked Examples: Lengths of Projections

Example 2

Find the length of the projection of AB on the x-axis, where A(2 , 3) and B(-4 , -1).

xy-6-6-4-4-2-22244666 unitsA(2,3)B(-4,-1)A'B'

Project both endpoints onto the x-axis:

A(2 , 3) → A'(2 , 0) , B(-4 , -1) → B'(-4 , 0)

The projection is A'B', so its length is

| -4 | + | 2 | = 6 length units

Equivalently | x₂ − x₁ | = | 2 − (−4) | = 6.

Example 3

Draw C(1 , 4) and D(5 , 1), then find the length of the projection of CD on the y-axis.

xy-6-6-4-4-2-22244663 unitsC(1,4)D(5,1)C'D'

C(1 , 4) → C'(0 , 4) , D(5 , 1) → D'(0 , 1)

Length of C'D' = | 4 | − | 1 | = 3 length units.

Example 4

Special cases: A(-4 , 3), B(-2 , 3) and then A(5 , 5), B(5 , -1).

xy-6-6-4-4-2-22244662ABA₂B₂

AB with A(-4 , 3), B(-2 , 3) is parallel to the x-axis, so its projection there has the same length: | −2 − (−4) | = 2 units.

A₂B₂ with A₂(5 , 5), B₂(5 , -1) is perpendicular to the x-axis, so its projection on the x-axis is the single point (5 , 0) — length 0, while its projection on the y-axis has length | 5 − (−1) | = 6 units.

Your turn

Try It Yourself

  • If A(-3 , 2) and B(4 , -4), find the length of the projection of AB on the x-axis and on the y-axis.
  • Find the projection length on the x-axis for A(-2 , 1), B(3 , 6) and for A(-3 , 4), B(2 , 2).
  • The projection of the point (a + 3 , a − 2) on the x-axis is the point itself. Find the point.
  • A point projects to (3 , 0) on the x-axis and to (0 , −2) on the y-axis. What is the point?