Handout 8 ( Vectors) MAT285
July –Nov.2009
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SELAMAT TINGGAL RAMADAN… SEMOGA DIRI KU MENJADI INSAN YANG LEBIH BERTAKWA…SEMOGA DAPAT
BERTEMU DGN MU LAGI…INSYAALLAH
Motivation
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Handout 8 ( Vectors) MAT285
July –Nov.2009
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Parametric Equations of Lines
a) The line in 2-space that passes through the point Px 0 , y 0  and is parallel to
the nonzero vector v = a, b  ai  bj has a parametric equations
x  x 0  at , y  y 0  bt
b) The line in 3-space that passes through the point Px 0 , y 0 z 0  and is parallel
to the nonzero vector v = a, b, c  ai  bj  ck has a parametric equations
...........................................................................................................
Example 1
Find the parametric equations of the line passing through (2,5) and parallel to
vector v  1,4
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Handout 8 ( Vectors) MAT285
July –Nov.2009
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Example 2
Find the parametric equations of the line passing through (1,2,-3) and parallel
to vector v  4i  5 j  7k.
Example 3
a) Find the parametric equations of the line L that passes through the points
P1(2,4,1) and P2 5,0,7 
b) Find the point of intersection of the line with the xy-plane.
Example 4 ( April 2007 )
Let A(3, 8, 4) and B(2, 9, 1) be two points in space. Find
i) the parametric equation of line that passes through the points A
and B.
ii) the intersection point between the line obtained in i) and yz-plane.
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Handout 8 ( Vectors) MAT285
July –Nov.2009
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Symmetric or Cartesian equation of a line in 3-space
Let L be the line that passes through the points x 0 , y 0 z 0  and is parallel to the
vector v  a, b, c where a, b and c are nonzero, then the symmetric equation or
cartesian equation of a line in 3-space is given by
.......................................................................................................
Example 5
Find the symmetric equation for the line passing through P(3,-1,2) and parallel
to the vector 6i-6j-2k
Example 6
Let
L1 : x  1  4t, y  5  4t, z  1  5t
L 2 : x  2  8t, y  4  3t, z  5  t
a) Is L1 parallel to L 2 ?
b) Do the line intersect? If the line intersect, find the point of intersection.
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Handout 8 ( Vectors) MAT285
July –Nov.2009
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Vector Equations of Lines
Let us define the following vectors:
r  x, y , r0  x 0 , y 0 and v  a, b
in 2 –space
or
r  x, y, z , r0  x 0 , y 0 , z 0 and v  a, b, c
in 3 –space
So, the vector equation of a line in 2-space or 3-space is given by
...................................................................................................
Example 7
Find the vector equation of the line in 3-space that passes through the points
P1(2,4,1) and P2 (5,0,7 ).
Ans: x, y, z  2,4,1)  t 3,4,8
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Handout 8 ( Vectors) MAT285
July –Nov.2009
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Planes Parallel to the Coordinate Planes
Planes Determined by a Point and a Normal vector
A plane in 3-space can be determined uniquely by specifying a point in the plane
and a vector perpendicular to the plane.
This vector is called .....................................
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Handout 8 ( Vectors) MAT285
July –Nov.2009
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Suppose we want to find an equation of the plane passing through
P0 x 0 .y 0 , z 0  and perpendicular to the vecor n  a, b, c
Define the following vectors:
r0  x 0 .y 0 , z 0 and r = .........................
So, n  r  r0   .......... .......... ...
or
a, b, c 
(1 )
=0
=.........................................................................................
( 2)
(3)
(3) This is called the point-normal form of the equation of a plane.
(1) and (2) are called the vector versions of the equation of a plane.
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Handout 8 ( Vectors) MAT285
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Example 8
Find the equation of the plane that passes through the point
P(-1,-1,2) and has  1,7,6 as the normal vector.
Theorem
If a,b,c and d are constants, and a,b and c are not all zero, then the graph of the
equation ax + by + cz + d = 0 is a plane that has the vector n = .................... as
a normal.
This equation is called the general form of a plane.
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Handout 8 ( Vectors) MAT285
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Example 9
Determine whether the planes
3x – 2y + z =4
6x -4y + 2z = 6
are parallel.
Example 10
Find an equation of the plane through the points
P11,2,1,P2 (2,3,1) and P3 (3,1,2) .
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Handout 8 ( Vectors) MAT285
July –Nov.2009
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Example 11
Determine whether the line
x = 3 + 8t,
y= 4 + 5t,
z=-3 –t
is parallel to the plane x-3y + 5z = 12.
Finding the angle between two intersecting plane
If n 1 is the vector normal to plane
L1 ,
n 2 is the vector normal to plane L 2 ,
Then the acute angle between the planes is given by .......................
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Handout 8 ( Vectors) MAT285
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Example 12
Find the acute angle of intersection between the two planes
2x – 4y + 4z = 6
6x + 2y – 3z = 4
Distance Between a Point in Space and a Plane
Theorem
The distance D between a point P0 x 0 , y 0 , z 0  and the plane
ax + by + cz + d = 0 is given by
...............................................................................................................
Example 13
Find the distance between the point (0,1,5) and the plane
3x +6y -2z – 5 = 0.
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Handout 8 ( Vectors) MAT285
July –Nov.2009
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Example 14
Find the distance between the given parallel planes.
-2x + y + z = 0
6x – 3y – 3z -5 = 0
Review Questions
Question 1 ( Apr 2009 )
Given A(0,0,1) and B(-1,1,3) are two points in space. Find
i)
Cartesian equation passing through point A
ii)
The intersection point between the line obtained in (i) and the plane
x+ 2y – 5z = 6
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Handout 8 ( Vectors) MAT285
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Question 2 (Oct 2008 )
Given a point A(5,0,2) and the line L: x= 3t + 7, y= -2t, z= t-1. Find
a)
the coordinateS where the line intersects the xy-plane
b)
the equation of the plane that contains point A and the line L.
Question 3 (April 2007 )
a)
Let A(2, -1, 3), B(3, 1, 2), C(0, 2, 1) and D(3,2, 4) be four points in three
dimensions.
b)
Find the equation of the plane containing points B, C and D.
Question 4 (Oct 2006 )
Find the intersection point and the angle of intersection between the lines
x  2 y 1 z  2
x 1


.
 y  4  z and
2
3
4
1
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Handout 8 ( Vectors) MAT285
July –Nov.2009
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Question 5 ( April 2006 )
Given two vectors u  2i  2 j  k and v  3i  4 j  k.
Find the unit vector perpendicular to the plane determined by these two vectors.
Question 6 ( Nov. 2005 )
A(2, 4, -2), B(-1, 2, -8), C(0, -5, 2) and D(1, 3, -1) are points in 3-space.
a) Find the equation of plane S containing points A, B and C.
b) Find the intersection point between plane S and line : x – 2= y = z – 1.
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