C. Geometry and Algebra of Vectors · Lines and Planes

Vector, Parametric and Symmetric Equations of a Line in Three-Space

C4.2About 70 minutes

Learning goals

I can explain why a line in three-space has no single scalar equation.
I can write vector, parametric and symmetric equations of a line in three-space from a point and a direction, or from two points.
I can test whether a point lies on a line and find where a line crosses the coordinate planes.
I can decide whether two lines in three-space are parallel, or the same line.

Words to know

three-space
The set of all points (x, y, z); also written ℝ³.

symmetric equations (three-space)
(x − x₀)/a = (y − y₀)/b = (z − z₀)/c, for a line through (x₀, y₀, z₀) with direction [a, b, c], a, b, c ≠ 0.

coordinate planes
The xy-plane (z = 0), the xz-plane (y = 0) and the yz-plane (x = 0).

direction vector
Any non-zero vector parallel to the line. All scalar multiples of it are also direction vectors.

Up off the page

A delivery drone lifts off from a rooftop and flies in a straight line: every second it moves 3 m east, 4 m north and 1 m up. To say where it is, you now need three coordinates (x, y, z). The good news: the vector equation from the last two lessons works with no changes at all. Only the vectors get one more component.

Remember from Vectors in Two-Space and Three-Space: a point in three-space is (x, y, z), the position vector of that point is [x, y, z], and |[a, b, c]| = √(a² + b² + c²).

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

In two-space, Ax + By + C = 0 is a line because there is only one direction perpendicular to the normal [A, B]. In three-space, a vector has a whole circle of directions perpendicular to it. So an equation like 2x − y + 3z = 6 describes a flat plane, not a line. (You will study planes in the next lesson, Linear Equations in Three Variables: Planes.)

So a line in three-space is described in one of two ways:

by vector or parametric equations (this lesson), or
as the intersection of two planes, using two scalar equations together (the lesson Two Planes and Their Line of Intersection).

In three-space a line has a whole wheel of perpendicular directions. So "perpendicular to n through P₀" picks out a plane, not a line, and no single scalar equation can describe a line in three-space.

Vector, parametric and symmetric equations in three-space

For a line through P₀(x₀, y₀, z₀) with direction vector m = [a, b, c] ≠ 0:

Vector: r = r₀ + tm, that is, [x, y, z] = [x₀, y₀, z₀] + t[a, b, c], t ∈ ℝ.
Parametric: x = x₀ + ta, y = y₀ + tb, z = z₀ + tc.
Symmetric (when a, b, c are all non-zero): (x − x₀)/a = (y − y₀)/b = (z − z₀)/c.

If one component of m is 0, that coordinate is constant. For direction [−3, 0, 2] through (3, 2, −1), the symmetric form is (x − 3)/(−3) = (z + 1)/2, y = 2.

Everything else carries over: two points give the direction m = AB; a point is on the line only if all three coordinates give the same t; lines are parallel when their directions are scalar multiples.

The line r = [2, 0, 1] + t[−1, 2, 1]. The arrow from t = 0 to t = 1 is the direction vector m = [−1, 2, 1]; each step of t adds one copy of it. The dashed drop lines show where each point sits above the xy-plane, which helps you see depth in a flat drawing.

Worked example

Example 1: The line through two points

Find vector, parametric and symmetric equations of the line through A(3, 2, −1) and B(0, 2, 1).

Direction: m = AB = [0 − 3, 2 − 2, 1 − (−1)] = [−3, 0, 2].
Vector: r = [3, 2, −1] + t[−3, 0, 2], t ∈ ℝ.
Parametric: x = 3 − 3t, y = 2, z = −1 + 2t.
Symmetric: the y-component of m is 0, so y stays 2: (x − 3)/(−3) = (z + 1)/2, y = 2.
Check B: t = 1 gives (0, 2, 1) ✓.

r = [3, 2, −1] + t[−3, 0, 2]; x = 3 − 3t, y = 2, z = −1 + 2t; (x − 3)/(−3) = (z + 1)/2, y = 2.

Worked example

Example 2: Is the point on the line?

Using the line from Example 1, decide whether P(−3, 2, 3) and Q(6, 2, −2) lie on it.

P: from x, 3 − 3t = −3, so t = 2. y = 2 ✓. z = −1 + 2(2) = 3 ✓. Every coordinate agrees with t = 2.
Q: from x, 3 − 3t = 6, so t = −1. y = 2 ✓. z = −1 + 2(−1) = −3, but Q has z = −2 ✗.

P is on the line (t = 2). Q is not.

Worked example

Example 3: Crossing the coordinate planes

Where does the line r = [1, −2, 4] + t[2, 1, −2] cross the xy-plane and the yz-plane?

The xy-plane is where z = 0: 4 − 2t = 0, so t = 2. Point: (1 + 4, −2 + 2, 0) = (5, 0, 0).
The yz-plane is where x = 0: 1 + 2t = 0, so t = −1/2. Point: (0, −2 − 1/2, 4 + 1) = (0, −2.5, 5).

It crosses the xy-plane at (5, 0, 0) (which is on the x-axis) and the yz-plane at (0, −2.5, 5).

Worked example

Example 4: From symmetric form, and a parallel line

The line L has symmetric equations (x − 2)/4 = (y + 1)/(−2) = z/3. Write its vector equation, and the vector equation of the line through (1, 0, 5) parallel to L.

Read the point from the numerators: x − 2, y + 1, z − 0, so P₀ = (2, −1, 0).
Read the direction from the denominators: m = [4, −2, 3].
L: r = [2, −1, 0] + t[4, −2, 3].
A parallel line uses the same direction: r = [1, 0, 5] + s[4, −2, 3].

L: r = [2, −1, 0] + t[4, −2, 3]; parallel line: r = [1, 0, 5] + s[4, −2, 3].

Just for fun

In two-space a line can get by with one equation. Move it up a dimension and it needs a point, a direction vector and a parameter, or two whole planes to hold it in place. Lines get high-maintenance in 3-D.

Hands-on activity

String in the corner of a room

You need: a ball of string, tape, a tape measure, and a corner of a room where two walls meet the floor. (Ask permission before you tape anything to a wall, and do not stand on furniture: choose points you can reach from the floor.)

Use the corner as the origin. Call one floor edge the x-axis, the other floor edge the y-axis, and the vertical edge the z-axis. Measure in metres.
Tape one end of the string to the floor at a point A, for example (1.0, 0.5, 0). Tape the other end to a wall at a point B you can reach, for example (0, 1.5, 1.2). Measure and record the actual coordinates of A and B.
Write the vector equation r = a + t(b − a).
Use t = 0.5 to predict the coordinates of the midpoint. Measure from the corner to check: distance along each floor edge, and height above the floor.
Use t = 0.25 to predict another point and check it.

What to look for: with A(1.0, 0.5, 0) and B(0, 1.5, 1.2), the direction is [−1.0, 1.0, 1.2] and the midpoint is (0.5, 1.0, 0.6). Your measured values should be within a few centimetres of your predictions (string sag adds a little error).

Your turn

Write vector and parametric equations of the line through (1, −4, 2) with direction [3, 0, −5].

Find a vector equation of the line through A(2, 1, 0) and B(5, −1, 4).

Write symmetric equations for the line in the previous question.

Is (8, −3, 8) on the line r = [2, 1, 0] + t[3, −2, 4]?

Find two more points on x = 1 + t, y = 2t, z = 3 − t.

Where does x = 2 + t, y = −1 + 3t, z = 6 − 2t meet the xy-plane?

Are r = [1, 0, 2] + t[2, −4, 6] and r = [3, −4, 8] + s[−1, 2, −3] the same line?

More practice: warm-up

What is a direction vector of r = [0, 1, 2] + t[5, −1, 3]?

Which point on r = [0, 1, 2] + t[5, −1, 3] has t = 1?

Write parametric equations for r = t[1, 1, 1].

Give a direction vector for the z-axis.

What does a single equation Ax + By + Cz + D = 0 describe in three-space?

More practice: on your own

Find a vector equation of the line through (4, 0, −1) parallel to (x − 1)/2 = (y + 3)/5 = (z − 2)/(−1).

Find where r = [2, −3, 6] + t[1, 3, −3] meets each coordinate plane.

Find a and b so that (a, b, 7) lies on x = 1 + 2t, y = −2 + t, z = 3 + 4t.

Are r = [1, 1, 1] + t[2, 0, −1] and r = [3, 2, 0] + s[−4, 0, 2] the same line, or parallel and distinct?

Find a vector equation of the line through the origin and the midpoint of A(2, 4, −6) and B(4, 0, 2).

Challenge

Show that (x − 1)/2 = (y + 1)/3 = (z − 4)/(−1) and (x − 5)/(−4) = (y − 5)/(−6) = (z − 2)/2 describe the same line.

The line L passes through P(1, 2, 3) with direction [2, −1, 2]. Find the two points on L that are exactly 6 units from P.

A drone is at (0, 0, 10) (metres) at t = 0 and moves with constant velocity [3, 4, −0.5] m/s. When and where does it reach the ground (z = 0), and how far has it flown?

Check your understanding

Quiz

Which of these describes a line in three-space?

Which parametric equations describe the line through (2, 0, −1) with direction [1, −3, 4]?

Which is a direction vector of the line through A(1, 2, 3) and B(4, 0, 5)?

Where does r = [1, 4, −2] + t[1, −2, 1] cross the xy-plane?

Which gives symmetric equations for the line through (3, −1, 0) with direction [2, 0, 5]?

Why is there no single scalar equation for a line in three-space?

For parents and teachers

Big idea (C4.2): lines in three-space use exactly the same vector equation r = r₀ + tm as in two-space, with a third component. What does not carry over is the scalar equation: one linear equation in x, y and z is a plane.

How to help: Use a room corner as the origin and ask your teen to describe a straight path (the string from a floor point to a wall point) with a vector equation. Then ask "Where is the halfway point?" and check with a tape measure.

Common difficulties:

Writing (y + 1)/0 in a symmetric equation. A zero component means that coordinate is constant.
Checking only two of the three coordinates when testing a point.
Reading the point from symmetric form with the wrong sign: (y + 1) means y₀ = −1.
Drawing 3-D diagrams: dashed drop lines to the xy-plane help show depth.