Projection Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Projection.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The image formed when points of a shape are mapped onto a lower-dimensional surface along parallel or converging rays.

A shadow cast on the ground is a projection—a 3D object mapped down to a 2D silhouette.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Projection loses a dimension while preserving some structural information.

Common stuck point: Different projections of the same 3D object can look very different depending on the angle of viewing.

Sense of Study hint: Try shining a flashlight on an object from different angles and trace the shadow each time to see how projections change.

Worked Examples

Example 1

medium
Find the orthogonal projection of point P(3, 7) onto the x-axis, the y-axis, and the line y = x.

Solution

  1. 1
    Step 1: Projection onto the x-axis: drop a perpendicular to the x-axis. The image is (3, 0).
  2. 2
    Step 2: Projection onto the y-axis: drop a perpendicular to the y-axis. The image is (0, 7).
  3. 3
    Step 3: Projection onto y = x: the projection of (x_0, y_0) onto y = x is \left(\dfrac{x_0+y_0}{2}, \dfrac{x_0+y_0}{2}\right) = \left(5, 5\right).

Answer

Onto x-axis: (3, 0); onto y-axis: (0, 7); onto y = x: (5, 5).
Orthogonal projection drops a perpendicular from the point to the target line. The projection onto y=x uses the formula where the projected point lies on the line at the foot of the perpendicular from P.

Example 2

hard
Vector \vec{a} = (3, 4) is projected onto vector \vec{b} = (1, 0). Find (a) the scalar projection and (b) the vector projection. Then find the projection onto \vec{c} = (1, 1)/\sqrt{2}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A 5 m flagpole stands vertically. At 3 pm the sun casts a horizontal shadow 12 m long. What is the length of the shadow (the projection of the pole onto the ground)?

Example 2

hard
Project the vector \vec{v} = (2, -1, 3) onto the unit vector \hat{u} = (0, 0, 1) (the z-axis direction). Interpret the result geometrically.
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Related Concepts

Dimension

Background Knowledge

These ideas may be useful before you work through the harder examples.

dimension