MCV4U · Calculus and Vectors · Sunway CIMP
Lines and Planes
A complete study of Topic H: the vector, parametric, symmetric and Cartesian equations of lines and planes; plotting and sketching in \(\R^3\); the geometry of intersections; distances from points to lines and planes; and the language of consistent and inconsistent systems. Work through the lessons in order, watch the two video lectures, then test yourself — every solution is typed and fully worked.
The whole topic in one picture: a line meets a plane at a single point when \(\vv{m} \cdot \vv{n} \neq 0\). When that dot product vanishes the line is parallel to the plane — and then everything depends on whether one point of the line satisfies the plane's equation.
Contents
The five lessons
At a glance
The forms you will keep coming back to
Almost everything in this topic is one of five equations, written in whichever form the question makes convenient. A line needs a point and a direction; a plane needs a point and either two directions or one normal. The cross product converts between those two views of a plane, and the dot product decides every question about how a line and a plane meet.
Keep this panel open beside the homework. Each lesson develops one row of it and shows where the form comes from — there is nothing here to memorise that is not derived somewhere on this site.
- Line, vector
- \(\vv{r} = \vv{r}_0 + t\vv{m}\)
- Line, symmetric
- \(\dfrac{x-x_0}{m_1} = \dfrac{y-y_0}{m_2} = \dfrac{z-z_0}{m_3}\)
- Plane, vector
- \(\vv{r} = \vv{r}_0 + s\vv{u} + t\vv{v}\)
- Plane, Cartesian
- \(Ax + By + Cz + D = 0\)
- Point to plane
- \(d = \dfrac{\abs{Ax_0 + By_0 + Cz_0 + D}}{\sqrt{A^2+B^2+C^2}}\)
Practise
Then test yourself
How the assessments are built
The four achievement categories
Knowledge
Reading a direction vector off a symmetric equation; stating a normal; quoting a distance formula correctly.
Application
Using a known method on a new set of numbers — finding an intersection, computing a distance, building a plane.
Thinking
Deciding which method a problem needs, classifying a system, or working backwards to find an unknown constant.
Communication
Explaining why in sentences: why one equation cannot describe a line in three-space; what an inconsistent system looks like.
Every one of the four assessments carries questions from all four categories, and so does the unit test.
Go further