MCV4U Lines in 2-Space: Notes and Practice

A robot lawnmower starts at the corner of a yard and drives in a straight line, $2$ m east and $1$ m north every second. You already know one way to describe its path, $y = m x + b$ . But that equation only tells you WHERE the path is. It cannot tell you where the mower is after $3$ seconds, and it falls apart for a vertical path. A vector description fixes both problems: start at a point, then take $t$ steps along a direction. That idea is the main tool of this unit, because it keeps working in 3-space, where $y = m x + b$ has no equivalent for a line.

The scalar equation and the normal vector

Every line in the $x y$ -plane can be written as $A x + B y + C = 0$ , where $A$ and $B$ are not both zero. This is the SCALAR (or standard) equation. Its coefficients hide a vector: $n = (A, B)$ is perpendicular to the line and is called a NORMAL vector. Here is why. Take two points $P_{1} (x_{1}, y_{1})$ and $P_{2} (x_{2}, y_{2})$ on the line. Both satisfy the equation, so subtracting $A x_{1} + B y_{1} + C = 0$ from $A x_{2} + B y_{2} + C = 0$ gives $A (x_{2} - x_{1}) + B (y_{2} - y_{1}) = 0$ . That says $n \cdot P_{1} P_{2} = 0$ : the normal is perpendicular to every segment of the line.

A x + B y + C = 0 n = (A, B) ⊥ line

Any non-zero multiple of (A, B) is also a normal to the same line.

VideoMr. Cioffi connects the scalar (Cartesian) equation Ax + By + C = 0 of a line to its normal vector.Watch on YouTube

The vector and parametric equations

Pick a point $P_{0} (x_{0}, y_{0})$ on the line and a non-zero vector $m = (m_{1}, m_{2})$ that points along it, called a DIRECTION vector. To reach any point $P$ on the line, go to $P_{0}$ and then move some multiple $t$ of $m$ . With $r_{0} = O P_{0}$ (the position vector of $P_{0}$ ) and $r = OP$ , this gives the VECTOR equation $r = r_{0} + t m$ , $t \in R$ . The number $t$ is the PARAMETER: $t = 0$ gives $P_{0}$ , $t = 1$ is one full step along $m$ , and negative values of $t$ go backward. Writing the two components separately gives the PARAMETRIC equations.

Fig 1A line through P₀ with direction vector m. The points for t = −1, 0, 1 and 2 are marked along it, with the vector equation r = r₀ + t m and the parametric equations x = x₀ + t m₁, y = y₀ + t m₂.

r = (x_{0}, y_{0}) + t (m_{1}, m_{2}) ⟺ x = x_{0} + t m_{1}, y = y_{0} + t m_{2}, t \in R

Normal and direction are perpendicular

In 2-space, if the normal is

n = (A, B)

, a direction vector is

m = (B, - A)

(- B, A)

: swap the components and change one sign. Check:

(A, B) \cdot (B, - A) = A B - B A = 0

. Going the other way, a line with direction

(m_{1}, m_{2})

has normal

(m_{2}, - m_{1})

. When

m_{1} \neq = 0

the slope is

\frac{m _{2}}{m _{1}}

, which equals

- \frac{A}{B}

when

B \neq = 0

You have	You want	How
Two points $A$ , $B$	Vector equation	Use $m = A B$ and either point as $r_{0}$
Vector or parametric	Scalar equation	Normal $(m_{2}, - m_{1})$ ; substitute the known point to find $C$ (or eliminate $t$ )
Scalar equation	Vector equation	Direction $(B, - A)$ ; find one point, for example by setting $x = 0$
Vector equation	Parametric equations	Write the $x$ and $y$ components separately

Converting between the three forms of a line in 2-space.

Equations of a line are not unique

The scalar equation is unique only up to a non-zero multiple (

x - 2 y - 4 = 0

and

2 x - 4 y - 8 = 0

are the same line). A vector equation is far less unique: any point of the line can serve as

r_{0}

and any non-zero multiple of

m

can serve as the direction. To test whether two vector equations describe the same line, check that the directions are parallel AND that a point of one line lies on the other.

VideoMr. Cioffi builds vector and parametric equations of a line in 2-space from a point and a direction vector.Watch on YouTube

Two lines in 2-space

Solving a system of two linear equations in $x$ and $y$ means finding the points that lie on both lines. Usually there is exactly one, the point of intersection. The exceptions happen when the normals are parallel, that is, when $(A_{1}, B_{1})$ is a multiple of $(A_{2}, B_{2})$ . Then the lines are parallel: if the whole equation is a multiple of the other, they are COINCIDENT (the same line, infinitely many solutions); if only the left side matches, they are PARALLEL AND DISTINCT and the system has no solution. For example, $2 x + 3 y = 6$ and $4 x + 6 y = 12$ are coincident, while $2 x + 3 y = 6$ and $4 x + 6 y = 5$ never meet.

Normals	Equations	Lines	Solutions
Not parallel	Any constants	Intersect once	Exactly one point
Parallel	One equation is a multiple of the other	Coincident	Infinitely many
Parallel	Left sides are multiples, constants are not	Parallel and distinct	None

The three possibilities for a system of two linear equations in two variables.

If the two lines are given in vector form, give them DIFFERENT parameters, set the $x$ components equal and the $y$ components equal, and solve the two equations for the two parameters. Substituting either parameter back gives the point.

Lines in 2-Space

Key things to remember

The scalar equation and the normal vector

The vector and parametric equations

Two lines in 2-space

Worked examples

Where marks get lost

Flashcards 7

Practice