Lines and Planes in Vector Form
Parametric and normal equations.
Key idea
Parametric and normal equations.
In depth
A line is r = p + td (point plus multiples of a direction). A plane is r = p + su + tv or n · (r − p) = 0 with normal n. These forms make intersections and distances computable.
Worked example
The line through (1, 2) with direction (3, 1): (x, y) = (1 + 3t, 2 + t).
Formula
Real-world uses
- Describing flight paths and the walls of a building in computer-aided design
Common mistake
Using a point where a direction vector is needed.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Lines and Planes in Vector Form?
Parametric and normal equations. It is a topic at the advanced level in Linear Algebra.
What is the formula for Lines and Planes in Vector Form?
The key formula is r = r₀ + t d.
Where is Lines and Planes in Vector Form used in real life?
Real-world examples: Describing flight paths and the walls of a building in computer-aided design.
What should I learn before Lines and Planes in Vector Form?
Make sure you are comfortable with Triple Product, the previous topic in Linear Algebra.
What is a common mistake with Lines and Planes in Vector Form?
Using a point where a direction vector is needed.
How can I practise Lines and Planes in Vector Form?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Lines and Planes in Vector Form?
Move on to Matrices: Definition next.
Related advanced topics in Linear Algebra
- Dot Product – Sum of component products.
- Angle Between Vectors – Cosine from the dot product.
- Orthogonality – Vectors are orthogonal when their dot product is 0.
- Projection – Component of one vector along another.
- Cross Product – A vector perpendicular to two vectors in 3D.
- Matrix Multiplication – Row-by-column products; not commutative.