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CHAPTER 3
BASIC CONCEPTS
What is scalar?
- a quantity that has only magnitude
What is vector?
- a quantity that has magnitude and direction
A vector can be represented by a directedline segment where the
i) length of the line segment represents
the magnitude of the vector
ii) direction of the line segment represents
the direction of the vector
VECTORS
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A vector can be written as QP
, or . The
order of the letters is important. QP
means
the vector is from P to Q, QP means vectoris from Q to P.
IfP 11 , yx is the initial point andQ 22 ,yx is the terminal point of a directedline segment, PQ, then component form of
vector v that represents PQ is
12
12
12121,,
yy
xxyyxxvv x
and the magnitude or the length ofv is
2
2
2
1
2
12
2
12
cc
yyxx
v
P
Q
Initial point
Terminal PointPQ
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Example:
Find the component form and length of the
vector v that has initial point 7,3 and
terminal point 5,2 .
Example:
Given 5,2v and 4,3w , find each of the
following vectors.
a) v21 b) vw c) wv 2
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Any vector that has magnitude of 1 unit iscalled unit vector.
Theorem
Ifa is a non-null vector and if a is a unit
vector having the same direction as a, then
a
aa
Example:
Find a unit vector in the direction of5,2
v and verify that it has length 1.
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STANDARD UNIT VECTORS
Three standard unit vectors are: i,j dan k
Vectors i,j and k can be written in
components form:
i = ,j = < 0,y, 0 >, k = < 0, 0,z >
and can interpreted as
a =
=xi +yj +zk
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The vector QP
with initial point
111 ,, zyxP and terminal point 222 ,, zyxQ has the standard representation
ki )()()( 121212 zzyyxxQP or
121212 ,, zzyyxxPQ
Example:
Let u be the vector with initial point (2,-5) and
terminal point (-1,3), and let jiv 2 . Writeeach of the following vectors as a linear
combination ofi andj.
a) u
b) vuw 32
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Example:
The vector v has a length of 3 and makes an
angle of 630
with the positivex-axis. Writev as a linear combination of the unit vectors i
andj.
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PROPERTIES OF VECTORS IN SPACE
Let 321 vvv ,,v and 321 www ,,w be
vectors in 3 dimensional space and kis aconstant.
1. wv if and only if
332211wvwvwv ,, .
2. The magnitude ofv is2
3
2
2
2
1 vvv v
3. The unit vector in the direction ofv is
v
,,
v
v 321 vvv
4. 332211 wvwvwv ,,wv
5. 321 kvkvkvk ,,v
6. Zero vector is denoted as 0 000 ,, .
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Example:
Express the vector PQ if it starts at point
),,( 856P and stops at point ),,( 937Q incomponents form.
Example:
Given that 213 ,,a , 461 ,,b . Find
(a) ba 3 (b) b
(c) a unit vector which is in the direction ofb.
(d) find the unit vector which has the same
direction as ba 3 .
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Example: (PARALLEL VECTORS)
Vector w has initial point (2,-1,3) and terminal
point (-4,7,5). Which of the following vectors isparallel to w?
Example: (COLLINEAR POINTS)
Determine whether the point 0,1,2,3,2,1 QP and 6,7,4 R lie on the same line.
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THE DOT PRODUCT
(THE SCALAR PRODUCT)
The scalar product between two vectors
v = 321
,, vvv and w = 321
,, www is
cosvwv w
where is the angle between v and w.
Example:Ifv = 2i-j+k, w = i+j+2k and the angle
between v and w is 60, find wv .
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Theorem : (The Dot Product)
If 321 vvv ,,v and 321 www ,,w , then the
scalar product wv is
wv 321 vvv ,, 321 www ,,
332211 wvwvwv
Example:
Given that 3,2,2 u , 1,8,5v and
2,3,4 w , find
(a) vu (b) wvu (c) wvu
(d) the angle between u and v
(e) the angle between v and w.
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Example:
LetA=(4,1,2),B=(3,4,5) and C=(5,3,1) are the
vertices of a triangle. Find the angle at vertexA.
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Theorem :(Angle between two vectors)
The nature of an angle
, between two vectorsu and v.
1. is an acute angle if and only if 0 vu
2. is an obtuse angle if and only if 0 vu
3. = 90
if and only if 0 vu
Example:Show that the given vectors are perpendicular to
each other.
(a) i andj
(b) 3i-7j+2k and 10i+4j-k
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Theorem (Properties of Dot Product)
Ifu,v, and w are nonzero vectors and kis ascalar,1. uvvu
2. wuvuwvu
3. vuvu kk
4.2
vvv
5. 0 u00u
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THE CROSS PRODUCTS
(VECTOR PRODUCTS)
The cross product (vector product) vu is a
vector perpendicular to u and v whose
direction is determined by the right hand rule
and whose length is determined by the
lengths ofu and v and the angle between
them.
u
vvu
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Definition : (Cross Product)
Ifu and v are nonzero vectors, and ( 0 )
is the angle between u and v, then
sinvuvu ,
Theorem (Properties of Cross Product)
The cross product obeys the laws
(a) 0uu
(b) uvvu
(c) wuvuwvu
(d) vuvuvu kkk
(e) u //v if and only if 0u v
(f) u 0u00
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Theorem
If kiu 321 uuu and kiv 321 vvv ,
then
ji
ki
122113312332
321
321
()()(
vu
vuvuvuvuvuvu
vvv
uuu
Example:
1. Given that 4,0,3u and 2,5,1 v , find
(a) vu (b) uv
2. Find two unit vectors that are perpendicular
to the vectors u 2i+2j-3k and v = i+3j+k.
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THE LENGTH OF THE CROSS
PRODUCT AS AN AREA
B Cu
A D
v
Area of a parallelogram vuvu sin
Area of triangle = vu2
1
Example:
(a) Find an area of a parallelogram that is
formed from vectors u = i +j - 3k and
v = -6j + 5k.
(b) Find an area of a triangle that is formed
from vectors u = i +j - 3k and
v = -6j + 5k.
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SCALAR TRIPLE PRODUCT
Theorem
If 111 ,, zyxa , 222 ,, zyxb and 333 ,, zyxc ,
then
333
222
111
zyx
zyx
zyx
cba
Note: cbacba
Example:
If a = 3i + 4j - k, b = -6j+5k and c = i+j-k,
evaluate
(a) cba (b) cba
(c) bac (d) cab
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In this section we use vectors to study lines in
three-dimensional space.
.
HOW LINES CAN BE DEFINED USING
VECTORS?
The most convenient way to describe a line in
space is to give a point on it and a nonzero vector
parallel to it.
LINES IN SPACE
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Suppose Lis a straight line that passes through
),,( 000 zyxP and is parallel to the vector
kiv cba .
Thus, a point ),,( zyxQ also lies on the line if
vtQP
.
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Say OP0r and OQr
0rr PQ
vrr t 0
vrr t 0
cbatzyxzyx ,,,,,, 000
Theorem (Parametric Equations for a Line)
The line through the point ),,( 000 zyxP and
parallel to the nonzero vector cba ,,A has the
parametric equations
atxx 0 , btyy 0 , ctzz 0 , (1)
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If we let 000 ,, zyx0R denote the position
vector of ),,( 000 zyxP and zyx ,,R the
position vector of the arbitrary point Q(x,y,z) on
the line, then we write equation (1) in the vector
form
ARR t0
Example:
Give the parametric equations for the line
through the point (6,4,3) and parallel to the
vector 7,0,2 .
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Theorem (Symmetric Equations for a line)
The line through the point ),,( 000 zyxP and
parallel to the nonzero vector cba ,,A has
the symmetrical equations
c
zz
b
yy
a
xx 000
Example:
Given that the symmetrical equations of a line in
space is2
4
4
3
3
12
zyx, find
(a) a point on the line.
(b) a vector that is parallel to the line.
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ANGLE BETWEEN TWO LINES
Consider two straight lines
c
zz
b
yy
a
xxl 1111 :
and
f
zz
e
yy
d
xxl 2222 :
The line 1l parallel to the vector kiu cba
and the line2
l parallel to the vector
kiv fed . Since the lines 1l and 2l are
parallel to the vectors u and v respectively, then
the angle, between the two lines is given by
vu
vu cos
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Example:
Find an acute angle between line
1l = i + 2j + t(2i
j + 2k)
and line
2l = 2ij + k + s(3i6j + 2k).
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INTERSECTION OF TWO LINES
In three dimensional coordinates (space), two
line can be in one of the three cases as shown
below
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Example:
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SHORTEST DISTANCE
a) Distance From A Point To A Line
sinQPd
where is the angle between v and vector
QP
.
since
sinQPQP
vv ,
we have the shortest distance ofQ fromLas
v
v QPd
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Example:
Find the shortest path from the point Q(2, 0, -2)
to the line
2
1
1
1
3
2:
zyxl
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Suppose that is a plane. Point ),,( 000 zyxP
and ),,( zyxQ lie on it. If ki cbaN is a
non-null vector perpendicular (ortoghonal) to ,
thenNis perpendicular to PQ .
Thus, 0
NPQ 0,,,, 000 cbazzyyxx
0)()()( 000 zzcyybxxa
PLANES IN SPACES
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Conclusion:
The equation of a plane can be determined if a
point on the plane and a vector orthogonal to the
plane are known.
Theorem (Equation of a Plane)
The plane through the point ),,( 000 zyxP and
with the nonzero normal vector cba ,,N has
the equation
Point-normal form:
0000 zzcyybxxa
Standard form:
dcybyax with 000 czbyaxd
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Example:
1. Give an equation for the plane through the
point (2, 3, 4) and perpendicular to the vector
4,5,6 .
2. Give an equation for the plane through the
point (4, 5, 1) and parallel to the vectors
A= 1,0,2 and B 4,1,0 .
3. Give parametric equations for the line through
the point (5, -3, 2) and perpendicular to the
plane 5726 zyx .
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INTERSECTION OF TWO PLANES
Intersection of two planes is a line. (L)
To obtain the equation of the intersecting line,
we need
1) a point on the lineL which is given by
solving the equations of the two planes.
2) a vector parallel to the lineL which is
21 NN
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If cbaNN ,,21 , then the equation of the
lineL in symmetrical form is
c
zz
b
yy
a
xx 000
Example:
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ANGLE BETWEEN TWO PLANES
Properties of two planes
(a) An angle between the crossing planes is an
angle between their normal vectors.
21
21cos
NN
NN
(b) Two planes are parallel if and only if their
normal vectors are parallel, 21 NN
(c) Two planes are orthogonal if and only if
021 NN .
Example:
Find the angle between plane 043
yx andplane 522 zyx .
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ANGLE BETWEEN A LINE AND A
PLANE
Let be the angle between the normal vector N to a
plane and the lineL. Then we have
Nv
Nv
cos
where v is vector parallel toL. Furthermore, if the
angle between the lineL and the plane , then
2
2
cos2
sinsin
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Nv
Nv sin
Example:
Find the angle between the plane 523 zyx
and the line 3
3
1
2
2
3
zyx.
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SHORTEST DISTANCE FROM A
POINT TO A PLANE(a) From a Point To a Plane
Theorem
The distance D between a point ),,( 111 zyxQ
and the plane dczbyax is
222
111
N
N
cba
dczbyaxPQD
Where ),,( 000 zyxP is any point on the plane.
),,( 111 zyxQ
,,( 000 zyxP
D
N
D
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Example
Find the distance D between the point (1, -4, -3)
and the plane 1632 zyx
Example
(a) Show that the line
1
1
23
1
zyx
is parallel to the plane 123 zyx .
(b) Find the distance from the line to the plane
in part (a)
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(b) Between two parallel planes
The distance between two parallel planes
1dczbyax and 2dczbyax is given by
222
21
cba
ddD
Example
Find the distance between two parallel planes
322 zyx and 7442 zyx .
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(c) Between two skewed lines
Remark:
Both formula can also be used to compute the
distance between 2 skewed lines.
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Example:
1. Find the shortest distance from P(1, -1, 2) to
the plane 3x7y +z = 5.
2. Find the shortest distance between the skewed
lines.
l1
:x = 1+2t,y = -1+ t,z = 2 + 4t
l2 :x = -2+4s,y = -3s ,z = -1+s
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