Lesson 2 of 5Lines & Planes

Unit 6 · Lesson 2 of 5

Lines in 3-Space

Why a line in 3-space has no single scalar equation, how to write its vector, parametric and symmetric equations from a point and a direction or from two points, and how to tell intersecting, parallel, coincident and skew lines apart.

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Key things to remember

1A single equation $A x + B y + C z + D = 0$ is a plane; a line in 3-space needs two equations (two intersecting planes).
2Vector: $r = r_{0} + t m$ ; parametric: $x = x_{0} + t m_{1}$ , $y = y_{0} + t m_{2}$ , $z = z_{0} + t m_{3}$ .
3Symmetric: $\frac{x - x _{0}}{m _{1}} = \frac{y - y _{0}}{m _{2}} = \frac{z - z _{0}}{m _{3}}$ (each $m_{i} \neq = 0$ ; a zero component makes that coordinate constant).
4Parallel directions: coincident or parallel and distinct. Non-parallel directions: intersecting or skew; the third equation decides.
5Angle between lines: $cos θ = \frac{∣ m _{1} \cdot m _{2} ∣}{∣ m _{1} ∣∣ m _{2} ∣};$ perpendicular when $m_{1} \cdot m_{2} = 0$ .

Two aircraft leave Toronto Pearson on straight climbing paths. On a flat map their tracks cross, so do the planes collide? Not necessarily: at the crossing point on the map, one plane may be thousands of metres above the other. In 3-space, two straight paths that are not parallel can still miss each other completely. Lines like that are called skew, and they have no equivalent in the $x y$ -plane. To work with them you need equations for lines in 3-space, and the vector form from the last lesson extends with almost no change.

Why there is no scalar equation of a line in 3-space

In 2-space, one equation $A x + B y + C = 0$ is a line. It is tempting to guess that $A x + B y + C z + D = 0$ is a line in 3-space, but it is not. One equation in three unknowns lets you choose two of the coordinates freely and then solve for the third, so its solutions spread out in two directions: they form a PLANE (the next lesson). A line has only one free direction, so it takes TWO equations to pin it down. Geometrically, a line in 3-space can be described as the line where two non-parallel planes cross. For example, the pair $y = 2$ and $2 x + 3 z = 3$ together describe a single line.

Vector, parametric and symmetric equations

r = (x_{0}, y_{0}, z_{0}) + t (m_{1}, m_{2}, m_{3}) x = x_{0} + t m_{1}, y = y_{0} + t m_{2}, z = z_{0} + t m_{3} \frac{x - x _{0}}{m _{1}} = \frac{y - y _{0}}{m _{2}} = \frac{z - z _{0}}{m _{3}}

Vector, parametric and symmetric equations of the line through (x₀, y₀, z₀) with direction (m₁, m₂, m₃), t a real number.

The vector and parametric equations work exactly as in 2-space, with a third component. The SYMMETRIC equations come from solving each parametric equation for $t$ and setting the results equal, so they need every $m_{i} \neq = 0$ . If a component of the direction is $0$ , that coordinate never changes: write it as a separate equation. The line through $(3, 2, - 1)$ with direction $(- 3, 0, 2)$ has symmetric equations $\frac{x - 3}{- 3} = \frac{z + 1}{2}$ , $y = 2$ .

Symmetric equations hide two planes

Each equals sign in the symmetric form is an equation in only two of the variables, so each is the scalar equation of a plane. Cross-multiplying

\frac{x - 3}{- 3} = \frac{z + 1}{2}

gives

2 x + 3 z - 3 = 0

, and together with

y = 2

it describes the line as the intersection of two planes.

VideoMr. Cioffi writes the vector, parametric and symmetric equations of a line in 3-space.Watch on YouTube

How two lines in 3-space can be related

Fig 1Four ways two lines in 3-space can be arranged: intersecting at one point, parallel and distinct, coincident, and skew (not parallel and never meeting, drawn along two parallel faces of a box).

Configuration	Direction vectors	Points in common
Coincident	Parallel (multiples)	Infinitely many: the same line
Parallel and distinct	Parallel (multiples)	None
Intersecting	Not parallel	Exactly one
Skew	Not parallel	None

Skew lines exist only in 3-space: in a plane, two non-parallel lines must meet.

Compare the direction vectors. If one is a multiple of the other, the lines are parallel: test whether a point of one line satisfies the other's equations. Yes means coincident, no means parallel and distinct.
If the directions are not parallel, give the lines different parameters ( $t$ and $s$ ) and set the $x$ , $y$ and $z$ components equal. That gives three equations in two unknowns.
Solve two of the equations for $t$ and $s$ .
Substitute both values into the third equation. If it is true, the lines intersect; substitute $t$ back to find the point. If it is false, the lines are skew.

Always use the third equation

Two equations in two unknowns usually have a solution, so finding

t

and

s

proves nothing yet. The third equation is the real test. Skipping it is the most common way to report a point of intersection for lines that are actually skew.

The angle between two lines is the angle between their direction vectors, found with the dot product from the vectors unit: $cos θ = \frac{m _{1} \cdot m _{2}}{∣ m _{1} ∣∣ m _{2} ∣} .$ Reversing one direction vector gives the supplementary angle, so the angle between lines is usually reported as the acute one (use $∣ m_{1} \cdot m_{2} ∣$ ). The lines are perpendicular when $m_{1} \cdot m_{2} = 0$ , and that is true even for skew lines, which can be perpendicular without touching.

VideoPatrick JMT shows how to decide whether two lines in 3-space coincide, are parallel, intersect or are skew.Watch on YouTube

Worked examples

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

Example 1

Represent the line through

(3, 2, - 1)

and

(0, 2, 1)

with a vector equation, parametric equations, symmetric equations, and the scalar equations of two intersecting planes.

Example 2

Determine whether

L_{1} : r = (1, - 2, 3) + t (2, 1, - 1)

and

L_{2} : r = (3, - 4, 7) + s (1, 2, - 3)

intersect. If they do, find the point.

Example 3

Show that

L_{1} : r = (1, 0, 2) + t (1, - 1, 1)

and

L_{2} : r = (0, 1, - 1) + s (2, 1, 0)

are skew.

Where marks get lost

Writing a "scalar equation of a line" such as $2 x + 3 y - z = 4$ in 3-space. One linear equation in $x$ , $y$ and $z$ is a plane.
Dividing by zero in the symmetric equations. When a direction component is $0$ , that coordinate is constant; write it separately, such as $y = 2$ .
Solving two of the three component equations and announcing an intersection without checking the third. For skew lines, two of the equations usually have a solution; only the third shows that the lines never meet.
Calling two lines that never meet "parallel". In 3-space they might be skew; parallel lines must have parallel direction vectors.
Using the same parameter $t$ for both lines, which forces the intersection to happen at the same parameter value on each line.

Flashcards 8

Vector equation of a line in 3-space

Parametric equations (3-space)

Symmetric equations

Why no scalar equation for a 3-D line?

+ 4 more

Practice

Why can't a line in 3-space be described by one scalar equation?

Find the

z

-coordinate of the point on

r = (2, - 1, 4) + t (3, 0, - 2)

where

t = - 2

The point

(10, k, 1)

lies on the line

x = 1 + 3 t

y = 2 - t

z = - 5 + 2 t

. Find

k

The line

\frac{x - 4}{2} = \frac{y + 1}{- 3} = \frac{z}{5}

passes through which point, with which direction vector?

How are

L_{1} : r = (1, 0, 2) + t (1, - 1, 1)

and

L_{3} : r = (2, 1, 0) + u (- 2, 2, - 2)

related?

Find the angle between the lines with direction vectors

(2, 1, - 1)

and

(1, 2, - 3)

, in degrees to one decimal place.

Match each test result for two lines in 3-space to the configuration.

Directions parallel, and a point of one line lies on the other

Directions parallel, and no point of one line lies on the other

Directions not parallel, and all three component equations hold

Directions not parallel, and the third component equation fails

What are two lines in 3-space called if they are not parallel and do not intersect? (one word)

Put the steps for deciding whether two non-parallel lines intersect in order.

1If it holds, find the point; if not, conclude the lines are skew
2Substitute both parameters into the remaining equation
3Solve two of the three equations for the two parameters
4Check that the direction vectors are not scalar multiples
5Give the lines different parameters and set the x, y and z components equal

In 3-space, what is the set of points satisfying both

x = 4

and

z = - 1

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