Practice Projection in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
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
hardFind the projection of onto .
Example 2
easyWhat do we call the 2D outline of a 3D object's shadow?
Example 3
mediumA 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
hardA line segment of length in 3D has its projection onto the -plane equal to a segment of length . What is the angle the segment makes with the plane?
Example 5
challengeA flat square of area 16 is tilted so it makes a angle with the projection plane. Find the area of its projection.
Example 6
easyA cube's shadow can be a square. What viewing/light direction gives this?
Example 7
easyA projection maps a 3D object onto a surface of what dimension?
Example 8
hardProject the vector onto the plane .
Example 9
mediumFind the orthogonal projection of point onto the -axis, the -axis, and the line .
Example 10
easyProject the point onto the -plane. What are the resulting coordinates?
Example 11
mediumA segment of length lies at angle to the projection plane. Find its projected length.
Example 12
easyA square is projected by parallel light rays at to its plane onto a perpendicular surface. The projection has the same _______ as the original.
Example 13
mediumA unit circle in 3D lies in a plane tilted from the projection plane. What is the projected shape and one of its axis lengths?
Example 14
easyA pencil held vertically casts a shadow. As the sun gets lower, what happens to the shadow's length?
Example 15
challengeIn a perspective projection from eye onto plane , find the image of the point .
Example 16
mediumWhy can a single projection (one view) be ambiguous about the original 3D object?
Example 17
easyA m flagpole stands vertically. At pm the sun casts a horizontal shadow m long. What is the length of the shadow (the projection of the pole onto the ground)?
Example 18
easyA cm segment lies in a plane parallel to the projection plane. What is the length of its orthographic projection?
Example 19
easyThe point is projected straight down onto the -plane. What is the image?
Example 20
hardProject the vector onto the unit vector (the -axis direction). Interpret the result geometrically.