Practice Projection in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

hard
Find the projection of a⃗=(1,2,3)\vec a = (1, 2, 3) onto b⃗=(2,2,1)\vec b = (2, 2, 1).

Example 2

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

Example 3

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 4

hard
A line segment of length 1313 in 3D has its projection onto the xyxy-plane equal to a segment of length 55. What is the angle the segment makes with the plane?

Example 5

challenge
A flat square of area 16 is tilted so it makes a 60∘60^\circ angle with the projection plane. Find the area of its projection.

Example 6

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

Example 7

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

Example 8

hard
Project the vector a⃗=(1,0,0)\vec a = (1, 0, 0) onto the plane x+y+z=0x + y + z = 0.

Example 9

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

Example 10

easy
Project the point (7,−2,5)(7, -2, 5) onto the yzyz-plane. What are the resulting coordinates?

Example 11

medium
A segment of length 2020 lies at angle 60∘60^\circ to the projection plane. Find its projected length.

Example 12

easy
A square is projected by parallel light rays at 30∘30^\circ to its plane onto a perpendicular surface. The projection has the same _______ as the original.

Example 13

medium
A unit circle in 3D lies in a plane tilted 45∘45^\circ from the projection plane. What is the projected shape and one of its axis lengths?

Example 14

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

Example 15

challenge
In a perspective projection from eye E=(0,0,10)E=(0,0,10) onto plane z=0z=0, find the image of the point P=(2,3,5)P=(2,3,5).

Example 16

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

Example 17

easy
A 55 m flagpole stands vertically. At 33 pm the sun casts a horizontal shadow 1212 m long. What is the length of the shadow (the projection of the pole onto the ground)?

Example 18

easy
A 1010 cm segment lies in a plane parallel to the projection plane. What is the length of its orthographic projection?

Example 19

easy
The point (3,4,7)(3, 4, 7) is projected straight down onto the xyxy-plane. What is the image?

Example 20

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

Dimension