Grade 12 - Lines and Planes
Explanation and Free Practice Resources
Represent lines in two- and three-dimensional coordinate geometry using vector, parametric, and symmetric equations, and represent planes in scalar form using normal vectors obtained from coefficients or cross products. Determine intersections by solving linear systems, including the point of intersection or parallel/coincident cases for two-dimensional lines, and intersecting, parallel, coincident, or skew relationships among three-dimensional lines and planes, with plane intersections expressed parametrically when appropriate. Calculate point-to-point, point-to-line, point-to-plane, and parallel-line or parallel-plane distances as minimum Euclidean lengths, identifying zero distance for intersecting objects and common perpendiculars for parallel or skew objects.
On this page:
- Master skills like “Write vector equations of lines”, “Write scalar equations of planes”, and “Determine normal vectors to planes” with unlimited, free practice problems
- Download (or modify) free worksheets on Lines and Planes you can use at home or in class with practice problems across all skills in this group including “Write parametric equations of lines”, “Determine intersections of two planes”, “Determine intersections of lines and planes”.
- Start working on adaptive units like “Lines and Vectors in 3d Space - Intro” or “Planes in 3D Space - Intro” that include these skills and will introduce and scaffold the skills carefully for maximum learning.
Lines and Planes: Individual Skills
Master 9 individual skills with free practice problems. Mouseover to see the skill's practice problems. Create a free account to track your progress.
Your results
- Calculate distances involving points, lines, and planes
- Determine intersections of lines and planes
- Determine intersections of two planes
- Determine normal vectors to planes
- Determine the point of intersection of two lines
- Determine whether lines are parallel, intersecting, or skew
- Write parametric equations of lines
- Write scalar equations of planes
- Write vector equations of lines
Practice 'Calculate distances involving points, lines, and planes'
Select any skill in this group by hovering over the list above. Unlimited free practice problems, automatically graded.
Hover a skill above to start practicing.
Download/Modify Free Grade 12 'Lines and Planes' Worksheet
Print this free 'Lines and Planes' worksheet to practice these skills away from the computer. Click 'Modify' to easily customize this worksheet to your exact needs.
This Worksheet Includes:
- Calculate distances involving points, lines, and planes
- Determine intersections of lines and planes
- Determine intersections of two planes
- Determine normal vectors to planes
- Determine the point of intersection of two lines
- Determine whether lines are parallel, intersecting, or skew
- Write parametric equations of lines
- Write scalar equations of planes
Units Teaching These Skills
These adaptive units introduce, teach, or reinforce skills in this group (Lines and Planes) through scaffolded problems that grow in difficulty as concepts are understood.