U12L1 Parametric and Vector Equations of a Line in a Plane
The equation of a line can be
thought of as the position of the
tip of your pen as time passes.
At time t, the x-coordinate and
y-coordinate can each be found
using their own linear equation.
The parametric equations of a straight line in a plane have the form
x = x0 + at
y = y0 + bt
where
(x, y) is the position vector of any point on the line
(x0, y0) is the position vector of some particular point on the line
(a, b) is a direction vector for the line
t ∈ R is the parameter
Each point on the line satis es BOTH equations.
Any vector that is parallel to the line may be used as its direction vector.
For example, the vector AB⃗ = OB⃗ − OA⃗ can be used as a direction
vector for the line that passes through the points A and B.
A line with direction vector (a, b) has slope
If a = 0, the line is vertical.
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U12L1 Parametric and Vector Equations
of a Line in a Plane
b
, provided a ≠ 0.
a
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Example
Determine the parametric equations of the line with direction vector (4, -2)
that passes through the point (-1, 3).
x = − 1 + 4t
y = 3 − 2t
where t ∈ R
To nd other points on the line, choose values for t.
t = 1 gives the point (3, 1)
t = 2 gives the point (7, -1)
Example
Find an equation for the line segment from A(-3, -1) to B(4, 2).
A direction vector is (4 - (-3), 2 - (-1)) = (7, 3)
A parametric equation for the line through A and B is
x = 4 + 7t
y = 2 + 3t where t ∈ R
To nd the restrictions on t, use the restrictions on x.
-3 ≤ x ≤ 4
-3 ≤ 4 + 7t ≤ 4
-7 ≤ 7t ≤ 0
-1 ≤ t ≤ 0
Therefore, the parametric equation for the line segment from A to B is
x = 4 + 7t
y = 2 + 3t where -1 ≤ t ≤ 0, t ∈ R
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U12L1 Parametric and Vector Equations
of a Line in a Plane
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The equation of a line can also
be expressed so that it gives
the position vector of the
points on it in terms of the
position vector of a particular
point on it.
The vector equation of a straight line in a plane has the form
r ⃗ = (x0, y0) + t(a, b)
where
r ⃗ = (x, y) is the position vector of any point on the line
(x0, y0) is the position vector of some particular point on the line
(a, b) is a direction vector for the line, and
t∈R
Example
State a vector equation of the line passing through P(-4, 6) and Q(2, 3).
PQ⃗ = (2 − (−4), 3 − 6) = (6, − 3) is a direction vector.
A vector equation is r ⃗ = (−4,6) + t(6, − 3), t ∈ R.
Vector equations are not unique.
For example, another vector equation is r ⃗ = (2,3) + t(2, − 1), t ∈ R.
U12L1 Parametric and Vector Equations
of a Line in a Plane
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