Vector equation of a line 1
A plane starts a journey at the
point (4,1) and moves each hour
æ5æ
along the vector æ æ.
æ2æ
a) Find the plane’s coordinate after
1 hour.
(9,3)
b) Find the plane’s coordinate after
2 hours.
(14,5)
c) Find the plane’s coordinate after
t hours.
æxæ æ4æ æ5æ
æ æ = æ æ+ tæ æ
Vector equation
æyæ æ1æ æ2æ
Parametric equation
4
(4,1)
2
5
10
A coordinate on the line.
x = 4 + 5t, y = 1 + 2t
The vector part of the line.
Vector equation of a line 2
Find the vector and parametric
equations of the straight line that
passes through A(x1,y1) and
B(x2,y2).
Find the vector that connects A to
B.
æx - x æ
1
AB = b - a = æ 2
æ
æy2 - y1æ
Vector equation will be:
æx æ æx - x æ
1
æ 1 æ+ tæ 2
æ
æy1 æ æy2 - y1 æ
Or using the B coordinate and the
vector BA.
æx æ æx - x æ
2
æ 2 æ+ sæ 1
æ
æy2 æ æy1 - y2 æ
Parametric equation will be:
(
)
(
)
+ s( x - x ), y = y + s( y - y )
x = x1 + t x2 - x1 , y = y1 + t y2 - y1
x = x2
1
2
2
1
or
2
Find the vector and parametric
equations of the straight line that
passes through A and B that have
position vectors æ1æ and æ6æ.
ææ
ææ
æ3æ
æ2æ
Vector equation:
æ6æ æ-5æ
æ1æ æ 5 æ
æ æ+ tæ æ or æ æ+ sæ æ
æ2æ æ 1 æ
æ3æ æ-1æ
Parametric equation:
x = 1 + 5t, y = 3 - t
x = 6 - 5s, y = 2 + s
or
Shortest distance problems
Find the shortest distance between Any point on the line will have the
the point P(12,4) and the straight coordinates:
line with the vector equation,
x = 1 + 3t, y = -3 + 5t
æ1 æ
æ æ+
æ-3æ
æ3æ
tæ æ
æ5æ
(the parametric equation of the line)
Find the vector QP.
The shortest distance from any
point to a line will make an angle
of 90o.
The vector QP and vector part of
the line meet at 90o.So the dot
product of the vectors will be 0.
33 - 9t + 35 - 25t = 0
æ11 - 3t æ æ3æ
æ
æ· æ æ = 0 34t = 68
æ 7 - 5t æ æ5æ
t =2
Q(x,y)
P(12,4)
(1,-3)
æ12æ æ 1 + 3t æ æ11 - 3t æ
QP = p - q = æ æ- æ
æ= æ
æ
æ 4 æ æ-3 + 5t æ æ 7 - 5t æ
Use this to find the magnitude of
QP.
æ11 - 3t æ æ 5 æ
æ
æ= æ æ
æ 7 - 5t æ æ-3æ
QP = 34
Intersecting lines
Vector lines can intersect, although Write down two vector line
they do not have to.
equations.
æ13æ æ-4æ
Example
æ2æ æ1æ
2 ships plan to meet at a buoy (B). S1 = æ æ+ tæ æ S2 = æ æ+ uæ æ
æ10æ æ 1 æ
3
3
æ
æ
æ
æ
Ship 1 starts at (2,3) and moves
along the vector æ1æ . Ship 2 starts
B has coordinates (x,y).
ææ
æ3æ
at (13,10) and moves along the
vector ææ-4ææ.
æ1 æ
B
S2
( x =)2 + t = 13 - 4u
( y =)3 + 3t = 10 + u
Solving this gives t=3, u=2.
Check this out with both
lines gives the coordinates:
B (5,12)
S1
Questions
1. Find the shortest distance from
the point P(20,3) to the straight
line with parametric equations,
2. Two ants set off to meet each
other at point M. The first ant
starts at (7,1) and the second ant
starts at (18,13). The ants are
x = 1 + 4t, y = -5 + 3t
moving along the vectors,
Make Q be the point on the line
æ1æ
æ-3æ
where QP and the line meet at 90o. A1 = æ æ, A2 = æ æ
æ2æ
æ-1æ
Find the vector QP.
a) Find the coordinates of M.
Find the value of t using the dot
M (12,11)
product.
b) Find the distance that the first
Find the numerical coordinates of
ant covers.
Q.
5 5
Find the magnitude of QP.
QP=5
c) Find the distance that the
second ant covers.
2 10