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: A projection maps a shape onto a lower-dimensional surface along rays — like a shadow flattening 3D to 2D.

Common stuck point: The procedure for projection is the easy part; the trap is thinking a projection keeps all the original information. Asking "Am I mapping a shape onto a lower-dimensional surface along rays?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I mapping a shape onto a lower-dimensional surface along rays?

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

Answer

Onto

x

-axis:

(3, 0)

; onto

y

-axis:

(0, 7)

; onto

y = x

(5, 5)

First step

Step 1: Projection onto the

x

-axis: drop a perpendicular to the

x

-axis. The image is

(3, 0)

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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}

Example 3

medium

Find the scalar projection of

\vec{a}=(4,3)

onto

\vec{b}=(2,0)

Example 4

medium

A regular hexagon of area

24

is tilted so its plane makes a

30^\circ

angle with the projection plane. Find the projection's area.

Example 5

medium

Find the projection of point

P(2, 5)

onto the line

y = 2x

Example 6

hard

Find the orthogonal projection of

\vec a = (3, 4, 12)

onto the plane

z = 0

. Then find the length of the rejection (component perpendicular to the plane).

Example 7

hard

Project the vector

\vec a = (1, 0, 0)

onto the plane

x + y + z = 0

Example 8

hard

Find the perpendicular distance from point

P(3, 0, 4)

to the line through the origin in direction

\vec d = (1, 2, 2)

Example 9

challenge

In a perspective projection from eye

E=(0,0,10)

onto plane

z=0

, find the image of the point

P=(2,3,5)

Example 10

challenge

A unit cube of volume

1

at the origin is projected orthogonally onto the plane

x+y+z=0

. Compute the area of the projection.

Practice Problems

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

Example 1

easy

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.

Example 3

easy

A ball's shadow on flat ground is what shape?

Example 4

easy

A projection maps a 3D object onto a surface of what dimension?

Example 5

easy

Does a projection always preserve the true lengths of an object?

Example 6

easy

A cube's shadow can be a square. What viewing/light direction gives this?

Example 7

easy

What do we call the 2D outline of a 3D object's shadow?

Example 8

easy

The point

(3, 4, 7)

is projected straight down onto the

xy

-plane. What is the image?

Example 9

easy

A flat world map is a projection of the round Earth. What does this cause?

Example 10

easy

A pencil held vertically casts a shadow. As the sun gets lower, what happens to the shadow's length?

Example 11

medium

4 \times 3

rectangle is tilted and its projection onto the floor is

4 \times 1.5

. Did the projection preserve area?

Example 12

medium

A cube can cast a hexagonal shadow. What light direction produces this?

Example 13

medium

The projection of

(5, 12, 8)

onto the

xz

-plane drops which coordinate, and what is the image?

Example 14

medium

Why can a single projection (one view) be ambiguous about the original 3D object?

Example 15

medium

A 1 m vertical stick casts a 2 m shadow. A nearby tree casts an 8 m shadow at the same time. How tall is the tree (using projection/similar triangles)?

Example 16

medium

An orthographic drawing shows front, top, and side views. Why use three projections instead of one?

Example 17

medium

A line segment of length 10 lies at

30^\circ

to the floor. What is the length of its projection (shadow) on the floor?

Example 18

medium

A circle is tilted relative to the projection plane. What shape is its projection?

Example 19

challenge

A flat square of area 16 is tilted so it makes a

60^\circ

angle with the projection plane. Find the area of its projection.

Example 20

challenge

Why must any flat map of the spherical Earth distort either area or angles (or both)?

Example 21

challenge

A 3D vector

(6, 8, 24)

is projected onto the

xy

-plane. Find the length of the projection and compare it to the original vector's length.

Example 22

challenge

A wireframe cube is rotated and projected to 2D, producing the familiar 'cube drawing' with a smaller square inside a larger one. Why do parallel edges of the cube stay parallel in this projection?

Example 23

easy

Find the orthogonal projection of point

P(-4, 9)

onto the

x

-axis.

Example 24

easy

Project the point

(7, -2, 5)

onto the

yz

-plane. What are the resulting coordinates?

Example 25

easy

A pole of height

h

stands vertically. The sun makes an angle

\theta

with the horizontal. Write the length of the projection (shadow) of the pole on flat ground.

Example 26

easy

10

cm segment lies in a plane parallel to the projection plane. What is the length of its orthographic projection?

Example 27

medium

Find the vector projection of

\vec a=(6,2)

onto

\vec b=(1,1)

Example 28

medium

A segment of length

20

lies at angle

60^\circ

to the projection plane. Find its projected length.

Example 29

medium

A unit circle in 3D lies in a plane tilted

45^\circ

from the projection plane. What is the projected shape and one of its axis lengths?

Example 30

medium

A vertical

2

m stick casts a shadow

3

m long. A nearby building casts a shadow

42

m. How tall is the building?

Example 31

medium

Compute the orthogonal projection of

\vec a = (5, 5, 0)

onto the line spanned by

\vec b = (1, 0, 0)

Example 32

medium

1

m square tile lies flat on the ground. The sun is at

45^\circ

above the horizon. What is the area of the tile's shadow on the ground?

Example 33

hard

Find the projection of

\vec a = (1, 2, 3)

onto

\vec b = (2, 2, 1)

Example 34

hard

A line segment of length

13

in 3D has its projection onto the

xy

-plane equal to a segment of length

5

. What is the angle the segment makes with the plane?

Example 35

hard

A unit cube has one space diagonal along the projection direction (orthographic). What polygon is its outline on the projection plane?

Example 36

hard

A square of area

50

in 3D has its projection onto a plane equal to a square of area

25

. What angle does the square's plane make with the projection plane?

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Related Concepts

Dimension

Background Knowledge

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

dimension