MCV4U Intersections of Lines and Planes: Notes and Practice

Key things to remember

1Line meets plane: substitute the parametric equations into the scalar equation and solve for

t

. If

n \cdot m = 0

the line is parallel to the plane or lies in it.

2Two non-parallel planes meet in a line with direction

n_{1} \times n_{2}

3Three planes meet at exactly one point if and only if

n_{1} \cdot (n_{2} \times n_{3}) \neq = 0

4Elimination giving

0 = 0

: infinitely many solutions (a line or a plane). Giving

0 = k

k \neq = 0

: no solution.

5Two equations in 2-space usually give a point; two equations in 3-space usually give a line.

Video games and animated films draw realistic light with ray tracing: for each pixel, the computer follows a straight ray from the camera and works out the first surface it hits. Each of those checks is a line-and-plane intersection, often repeated millions of times for a single frame. The same algebra answers older questions too. A system of three linear equations in three unknowns, which you might have solved by elimination before, is really three planes, and its solution is wherever those planes meet. This lesson connects each algebraic outcome to a geometric picture.

Number of equations	Solutions in 2-space (x, y)	Solutions in 3-space (x, y, z)
One linear equation	A line	A plane
Two equations (not parallel)	A point	A line
Three equations (in general position)	Usually no common point	A point

In the typical case each new independent equation removes one free direction.

A line and a plane

Write the line in parametric form and substitute $x$ , $y$ and $z$ into the scalar equation of the plane. Everything collapses to one linear equation in $t$ . Solve it and substitute $t$ back into the line to get the point. Whether that equation has a solution depends on $n \cdot m$ , the dot product of the plane's normal with the line's direction: if it is $0$ , the line runs parallel to the plane.

Test	Equation in t	Geometry
$n \cdot m \neq = 0$	$k t = c$ with $k \neq = 0$ : one value of $t$	The line crosses the plane at one point
$n \cdot m = 0$ , point of line not on plane	$0 t = c$ with $c \neq = 0$ : no solution	Parallel to the plane, never meets it
$n \cdot m = 0$ , point of line on plane	$0 t = 0$ : every $t$ works	The line lies in the plane

The three ways a line and a plane can be arranged.

VideoFinds the point where a line meets a plane, working from the line's parametric equations and the plane's equation.Watch on YouTube

Two planes

Fig 1Two planes can intersect in a line, be parallel and distinct, or be coincident. In the last two cases their normals are parallel.

If the normals are parallel, the planes are parallel and distinct (no solution) or coincident (one equation is a multiple of the other). Otherwise the planes meet in a line. That line lies in both planes, so its direction is perpendicular to both normals: $m = n_{1} \times n_{2}$ . For a point on it, set one variable to a convenient value (often $z = 0$ ) and solve the remaining two equations. Pick a variable that really can take that value: if $m$ has a zero $z$ component, the line may never reach $z = 0$ , so set $x$ or $y$ instead. Alternatively, eliminate one variable and let a remaining variable be the parameter $t$ .

VideoKrista King finds parametric equations for the line where two planes meet.Watch on YouTube

Three planes

Fig 2The main configurations of three planes: meeting at one point; meeting in a common line, like the pages of a book; a triangular prism, where each pair meets in a line but there is no common point; two parallel planes cut by a third; and three parallel planes. Coincident cases are noted too.

Configuration	Normals	Solutions of the system
Meet at one point	Not coplanar: $n_{1} \cdot (n_{2} \times n_{3}) \neq = 0$	Exactly one $(x, y, z)$
Meet in a common line (a sheaf)	Coplanar, none parallel	Infinitely many, one parameter
Triangular prism	Coplanar, none parallel	None
Two parallel planes cut by a third	Two parallel	None
Three parallel distinct planes	All parallel	None
Two coincident planes cut by a third	Two parallel	Infinitely many, one parameter (a line)
Three coincident planes	All parallel	Infinitely many, two parameters (a plane)

Organizing three planes by how they intersect: at a point, along a line, in a plane, or not at all.

The triple product test

Let

a

b

and

c

be normals to the three planes. If

a \cdot (b \times c) \neq = 0

, the normals do not lie in one plane, and the three planes meet at exactly one point. If

a \cdot (b \times c) = 0

, the normals are coplanar and there is NOT a single point: the system has either no solution or infinitely many, and elimination tells you which.

To solve three equations by elimination, use two different pairs of equations to eliminate the same variable, leaving two equations in two unknowns. Solve those, then back-substitute. Watch for two special results. If an equation reduces to $0 = 0$ , one equation adds nothing new: the planes share a line (or more), and you describe the solutions with a parameter. If an equation reduces to something false such as $0 = 2$ , the system is inconsistent and there is no common point.

Loading interactive…

Enter the equations of two or three planes and rotate the 3-D view to see whether they meet at a point, along a line or not at all, with the solution of the system shown.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

Find the point where the line

r = (2, - 1, 3) + t (1, 2, - 1)

meets the plane

3 x - y + 2 z - 15 = 0

Example 2

Find a vector equation of the line of intersection of the planes

x + 2 y - z = 4

and

2 x - y + z = 3

Example 3

Solve the system

x + y + z = 6

2 x - y + z = 3

x + 2 y - z = 2

and describe it geometrically.

Example 4

Let

P_{1} : x + y + z = 1

and

P_{2} : x - y + z = 0

. Find a plane

P_{3}

that meets

P_{1}

and

P_{2}

at a single point, and a plane

P_{4}

that meets them in more than one point.

Where marks get lost

Reading

n \cdot m = 0

as "the line is perpendicular to the plane". It means the line is perpendicular to the NORMAL, so the line is parallel to the plane (or lies in it).

Concluding "no solution" from

n_{1} \cdot (n_{2} \times n_{3}) = 0

. A zero triple product rules out a single point, but the planes might still share a whole line.

Using one of the normals as the direction of the line where two planes meet. The direction is

n_{1} \times n_{2}

, perpendicular to both normals.

Setting

z = 0

to find a point on a line of intersection whose direction has a zero

z

component. That line is parallel to the

x y

-plane and may never reach

z = 0

; set another variable instead.

Getting

0 = 0

during elimination and writing "no solution". A true statement like

0 = 0

means infinitely many solutions; a false one like

0 = 2

means none.

Key things to remember

Worked examples

Where marks get lost

Flashcards 8

Practice