C. Geometry and Algebra of Vectors · Lines and Planes

Vector and Parametric Equations of a Line in Two-Space

C4.1About 70 minutes

Learning goals

I can write a vector equation r = r₀ + tm of a line from a point and a direction vector, or from two points.
I can write parametric equations of a line and use them to find points on it.
I can test whether a point lies on a line given in vector or parametric form.
I can decide whether two vector equations describe the same line.

Words to know

vector equation of a line
r = r₀ + tm, t ∈ ℝ, where r₀ is the position vector of a point on the line and m is a direction vector.

position vector
The vector from the origin to a point. The point (3, 2) has position vector [3, 2].

parametric equations
Equations that give each coordinate separately in terms of a parameter, such as x = x₀ + tm₁, y = y₀ + tm₂.

parameter
A variable (here t or s) that runs through all real numbers; each value gives one point.

collinear
Lying on the same straight line.

Where will you be, and when?

A snowmobile leaves a trailhead at the point (2, 1) on a map grid (units in kilometres) and travels in a straight line. Every hour, it moves 6 km east and 8 km north. Where is it after 1 hour? (8, 9). After 2 hours? (14, 17). After half an hour? (5, 5).

You just used a parameter: the time t. The position is "start point + t × (movement per hour)". That is the idea behind the vector equation of a line. It is the form that works in every dimension, which is why it is the main tool for the rest of this unit.

The vector equation of a line

To pin down a line you need two things: one point on it and its direction.

Let r₀ be the position vector of a known point P₀ (the vector from the origin O to P₀).
Let m be a non-zero direction vector parallel to the line.
Let r be the position vector of any point P(x, y) on the line.

Then r = r₀ + t m, t ∈ ℝ.

Each real number t gives one point. t = 0 gives P₀ itself, positive t moves along m, negative t moves the other way. As t runs through all real numbers, the tips of r trace out the whole line.

In two-space, with r₀ = [x₀, y₀] and m = [m₁, m₂], this reads

[x, y] = [x₀, y₀] + t[m₁, m₂].

(Some textbooks write vectors with round brackets, like (m₁, m₂). In this unit, square brackets are vectors and round brackets are points.)

The line r = [1, 1] + t[2, 1]. Start at the tip of r₀ (t = 0) and add t copies of the direction vector m. Each value of t lands on one point of the line; t = 2 gives the point (5, 3).

Parametric equations

Split the vector equation into its components. Equal vectors have equal components, so

x = x₀ + t m₁, y = y₀ + t m₂, t ∈ ℝ.

These are the parametric equations of the line. They say the same thing as the vector equation, one coordinate at a time. For the line in the diagram, x = 1 + 2t and y = 1 + t.

Good to know:

The equations are not unique. Any point on the line can serve as P₀, and any non-zero multiple of m is still a direction vector.
To find the line through two points A and B, use m = AB = B − A (or any multiple of it).
To test whether a point lies on the line, solve for t from each parametric equation. The point is on the line only if every component gives the same t.
If you restrict t to 0 ≤ t ≤ 1 in r = a + t(b − a), you get just the segment from A to B. Then t = 1/2 is the midpoint.

Two different vector equations for the same line. They use different points (A and B) and different direction vectors, but [−4, −2] = −2[2, 1], so the directions are parallel, and B is on the first line (t = 2).

Worked example

Example 1: Writing the equations and making points

Write a vector equation and parametric equations of the line through A(2, −1) with direction vector m = [3, 4]. Find the points where t = −1, t = 0 and t = 2.

Vector equation: r = [2, −1] + t[3, 4], t ∈ ℝ.
Parametric equations: x = 2 + 3t, y = −1 + 4t.
t = −1: x = 2 − 3 = −1, y = −1 − 4 = −5, so (−1, −5).
t = 0: (2, −1), the starting point.
t = 2: x = 2 + 6 = 8, y = −1 + 8 = 7, so (8, 7).

r = [2, −1] + t[3, 4]; x = 2 + 3t, y = −1 + 4t. Points: (−1, −5), (2, −1), (8, 7).

Worked example

Example 2: The line through two points, and testing points

Find a vector equation of the line through A(−1, 4) and B(3, 2). Then decide whether C(7, 0) and D(5, 2) are on the line.

Direction: AB = [3 − (−1), 2 − 4] = [4, −2]. A simpler parallel vector is [2, −1].
Vector equation: r = [−1, 4] + t[2, −1]. Parametric: x = −1 + 2t, y = 4 − t.
Test C(7, 0): from x, −1 + 2t = 7 gives t = 4. From y, 4 − t = 0 gives t = 4. Same t, so C is on the line.
Test D(5, 2): from x, t = 3. From y, 4 − t = 2 gives t = 2. Different values, so D is not on the line.

r = [−1, 4] + t[2, −1]. C is on the line (t = 4); D is not.

Worked example

Example 3: Same line or not?

Do r = [1, 2] + s[−2, 6] and r = [−3, 14] + t[1, −3] describe the same line?

Compare directions: [−2, 6] = −2[1, −3]. The directions are parallel, so the lines are parallel or the same.
Check one point: is (−3, 14) on the first line? From x: 1 − 2s = −3 gives s = 2. From y: 2 + 6(2) = 14 ✓.
Parallel lines that share a point are the same line.

Yes, they describe the same line.

Worked example

Example 4: Parameter as time

The snowmobile from the start of the lesson starts at (2, 1) with velocity [6, 8] km/h. Write its position at time t, find its speed, and find when and where it crosses a north-south trail along x = 14.

Position: r = [2, 1] + t[6, 8], or x = 2 + 6t, y = 1 + 8t, with t in hours and t ≥ 0.
Speed is the magnitude of the velocity: |[6, 8]| = √(36 + 64) = 10 km/h.
Crossing x = 14: 2 + 6t = 14, so t = 2.
Then y = 1 + 8(2) = 17.

Speed 10 km/h; it crosses the trail after 2 h, at (14, 17).

Just for fun

Parametric equations give a line a schedule as well as a route. Finally, a line that can tell you when it will arrive. (Unlike some buses.)

Hands-on activity

Walk the parameter

You need: a sheet of graph paper (or a tiled floor with square tiles), a coin or button as a marker, and a pencil.

Choose a start point, say P₀(1, 1), and a direction vector, say m = [2, 1].
Put the coin on P₀ and call it t = 0. Move it by m (2 right, 1 up) and mark t = 1. Repeat for t = 2 and t = 3. Then go back to P₀ and move by −m to mark t = −1.
Lay a ruler along your marks. They should all be on one straight line.
Now try a half step: where is t = 0.5? Check with the parametric equations x = 1 + 2t, y = 1 + t.
Repeat with direction 2m = [4, 2] from the same start. Do you get the same line? Which values of t give the same points as before?

What to look for: all marks are collinear; t = 0.5 is at (2, 1.5); with [4, 2] you get the same line, and each point appears at half the earlier t-value.

Your turn

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

Find three points on the line r = [0, 2] + t[3, −1].

Find a vector equation of the line through A(1, 1) and B(4, 7).

Is (−2, 9) on the line x = 1 − t, y = 3 + 2t?

Is (5, 0) on the line r = [−1, 4] + t[2, −1]?

State a point and a direction vector for the line x = 7 + 2t, y = −5t.

Write a vector equation of the horizontal line through (3, −2).

More practice: warm-up

For r = [5, 1] + t[2, 3], which point corresponds to t = 0?

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

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

Write parametric equations for r = [0, 0] + t[1, 4].

Give a direction vector for the line through (0, 0) and (2, 5).

More practice: on your own

Find the x- and y-intercepts of the line x = 6 − 3t, y = 1 + t.

Do r = [2, 5] + t[4, −6] and r = [0, 8] + s[−2, 3] describe the same line?

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

Find a vector equation of the line through (3, −1) that is parallel to the line through (0, 0) and (−2, 7).

Write a vector equation for the segment from A(−2, 1) to B(6, 5), and find the point one quarter of the way from A to B.

Challenge

Snowmobile A follows r = [0, 0] + t[4, 3] and snowmobile B follows r = [10, 0] + t[−2, 6], with distances in km and t in hours after noon. Do their paths cross? Do the snowmobiles collide?

Find k so that the point (k, 10) is on the line x = 3 + 2t, y = −2 + 4t.

Show that the lines r = [1, 3] + t[2, −1] and r = [4, 0] + s[1, 2] are perpendicular, and find their point of intersection.

Check your understanding

Quiz

Which is a vector equation of the line through (2, −1) with direction vector [3, 4]?

For x = 1 + 2t and y = 5 − t, which point corresponds to t = 3?

Which vector is a direction vector of the line through (1, 5) and (4, −1)?

In the vector equation r = r₀ + tm, what is m?

Is (4, 7) on the line r = [1, 1] + t[1, 2]?

For r = a + t(b − a), which values of t give exactly the segment from A to B?

For parents and teachers

Big idea (C4.1): a line is "a point plus any multiple of a direction". In symbols, r = r₀ + tm. Splitting it into coordinates gives parametric equations. This form will carry straight over to three dimensions in a later lesson of this unit.

How to help: Hand your teen a point and a direction and ask for three points on the line (t = 0, 1, −1). Then name a point and ask, "Is it on the line? Show me the value of t." The key habit is checking that every coordinate gives the same t.

Common difficulties:

Using the two given points as "point and direction" (for example writing [1, 5] + t[4, −1] for the line through (1, 5) and (4, −1)). The direction must be the difference B − A.
Thinking two equations are different lines because they look different. Check: parallel directions, plus one shared point.
Using the same letter t for two different lines when finding where they cross. Use t for one and s for the other.
Forgetting "t ∈ ℝ" (or the restriction 0 ≤ t ≤ 1 for a segment).