March 6, 2020
LINES IN SPACE
YOU WILL BE ABLE TO
• Parametric
• Vector
• Cartesian
equations of lines
• Angle between two lines
• Parallel & Perpendicular Lines
• Skew Lines
• Intersecting Lines
• Application of Lines to Motion
March 6, 2020
YOU WILL BE ABLE TO
PARAMETRIC EQUATION
• A straight line L that passes
through the point
• A(x0,y0,z0) and parallel to the
vector v=ai+bj+ck as shown in
the diagram.
• If L is the line that passes
through A and is parallel to the
non-zero vector v, then L
consists of all the points
M(x,y,z) for which the vector
AM is parallel to v.
March 6, 2020
• AM=tv, where t is a scaler.
(x-x0,y-y0,z-z0)=t(a,b,c)=(ta,tb,tc).
1) Parametric Equation of a Line:
(x-x0,y-y0,z-z0)=t(a,b,c)
March 6, 2020
EXAMPLE:
a) Find parametric equations of the line through A(1,-2,3)
and parallel to v=5i+4j-6k.
b) Find parametric equations of the line through the points
A(1,-2,3) and B(2,4,-2).
VECTOR EQUATION OF A LINE
AM = tv
AM=OM - OA
tv=r-r0
March 6, 2020
2) Vector Equation of a Line:
(x-x0,y-y0,z-z0)=t(a,b,c)
(x, y, z) = (x0, y0, z0) + t(a,b,c)
EXAMPLE:
Find a vector equation of the line that contains (-1,3,0)
and is parallel to v=3i-2j+k.
March 6, 2020
EXAMPLE:
Find a vector equation of the line passing through
A(2,7) and B(6,2).
EXAMPLE:
Find the parametric equations of the line parallel to v=(1,-4,2)
which passes through P(-3,2,-1).
March 6, 2020
EXAMPLE:
Find the parametric equations for the line through P(-1,1,3)
and parallel to the vector v=2i+3j-k .
CARTESIAN EQUATIONS OF LINES
The vector v is : v=(a,b,c) and
the point A is: A(x0,y0,z0)
March 6, 2020
EXAMPLE:
Find the Cartesian equations of the line l that passes through the
point P(1,2,-3) and is parallel to the vector v=(-2,1,-4).
EXAMPLE:
Find the Cartesian equations of the line through
A(3,-7,4) and B(1,-4,-1).
March 6, 2020
EXAMPLE:
Let L be the line with Cartesian equations
Find a set of parametric equations for L.
EXAMPLE:
Find the equation of the line that passes through the point
P(1,0,-3) and which is parallel to the line
March 6, 2020
EXAMPLE:
Find the parametric and Cartesian equation of the line through the
point P(-1,0,5) which is parallel to the line
EXAMPLE:
Given the points A(1,3,-1) and B(2,5,0), represent the line (AB) by
a. a vector equation
b. parametric equation
c. cartesian equation
March 6, 2020
EXAMPLE:
A line l has Cartesian equation
Find the coordinates of one of the points on line l and a
vector parallel to l.
March 6, 2020
EXAMPLE: Find the cosine of the angle between the lines
March 6, 2020
EXAMPLE:
are parallel lines. Find p and q.
March 6, 2020
EXAMPLE: The lines
Find p.
EXAMPLE:
Show that the lines l1: x=2+3t, y=-1+t, z=-2t
and
l2 : x=-3-k, y=2+5k, z=-2+k are perpendicular to each other.
March 6, 2020
SKEW LINES
EXAMPLE:
The lines L1 and L2 have the following equations:
L1: x=1+4t, y=5-4t, z=-1+5t
L2: x=2+8s, y=4-3s, z=5+s
Show that the lines are skew.
March 6, 2020
INTERSECTING LINES
The lines L1 and L2 have the following equations:
L1: x=1+2t, y=3-4t, z=-2+4t
L2: x=4+3s, y=4+s, z=-4-2s
Show that the lines intersect.
APPLICATION OF LINES TO MOTION
r(t) = r0 + tv
= r0 +
initial
position
t
time
.
|v|
speed
.
direction
March 6, 2020
EXAMPLE:
A model plane is to fly directly from a platform at a reference point
(2,1,1) toward a point (5,5,6) at a speed of 60 m/min. What is the
position of the plane ( to the nearest metre) after 10 minutes ?
EXAMPLE:
An object is moving in the plane of an appropriately fitted
coordinate systemsuch that its position is given by
r=(3,1)+t(-2,3), where t stands for time in hours after start and
distances are measured in km.
a) Find the initial position of the object.
b)Show the position of the object on a graph at start, 1 hour and
3 hours after start.
c) Find the velocity and speed of the object.
March 6, 2020
EXAMPLE:
At 12:00 midday a plane A is passing in the vicinity of an
airport at a height of 12 km and a speed of 800 km/h. The
direction of the plane is (4,3,0). [Consider that (1,0,0) is a
displacement of 1 km due east (0,1,0) due north, and (0,0,1) is
an altitude of 1 km.]
a) Using the airport as the origin, find the position vector r of the
plane t hours after midday.
b) Find the position of the plane 1 hour after midday.
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c) another plane B is heading towards the airport with velocity vector
(-300, -400, 0) from a loation (600, 480, 12). Is there a danger of
collision ?
EXAMPLE:
March 6, 2020
HW PAGE:
Pages 667-668-669-670
#1-3-4-5-7a-8-9-15-18-23-24
IB QUESTION BANK
PAPER 1
TURN THE PAGE
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