Date post: | 02-Jun-2018 |
Category: |
Documents |
Upload: | daphne-liew |
View: | 221 times |
Download: | 0 times |
of 27
8/11/2019 C1L1 Vectors and the Three Dimensional Space
1/27
UTP/JBJ 1
Vectors and the ThreeDimensional Space
Chapter 1
Lesson 1
8/11/2019 C1L1 Vectors and the Three Dimensional Space
2/27
UTP/JBJ 2
Learning Outcomes
At the end of the lesson you should be ableto:
1. Identify a vector in 3- space from 2-space
and 1-space.2. Find the distance between points.
3. Derive the equation of sphere.
4. Define vector.5. Perform the arithmetic of vectors.
6. Perform Dot and Cross product
8/11/2019 C1L1 Vectors and the Three Dimensional Space
3/27
UTP/JBJ 3
Rectangular Coordinate Systems One way to identify points in a plane is to use a
Cartesian coordinate system.
3- space: x, y, and zaxes 2- space: xand yaxes produces a Plane
1-space : xaxes, a line
z
y
x
o
y
xo
xo
8/11/2019 C1L1 Vectors and the Three Dimensional Space
4/27
8/11/2019 C1L1 Vectors and the Three Dimensional Space
5/27
Mid-point The coordinate of the center between points
A(a,b) and B(c,d) is given by
The coordinate of the point divides points
A(a,b) and B(c,d) in a ratio m:n is given by
UTP/JBJ 5
2,
2
dbcaM
nm
dmbn
nm
cman,
8/11/2019 C1L1 Vectors and the Three Dimensional Space
6/27
UTP/JBJ 6
Distance (d) in 3-space
22 ba
222 cbad
),,( 1111 zyxP
d c
b
a
y
),,( 2222 zyxP
2
12
2
12
2
12 )()()( zzyyxxd
czzbyyaxx 121212 ;;
8/11/2019 C1L1 Vectors and the Three Dimensional Space
7/27
UTP/JBJ 7
SPHERE
222
2222 )0,0,0(,
zyxr
Czyxr
z
y
x
r
C(h,k,l)
P(x,y,z)
222 )()()( lzkyhxr
8/11/2019 C1L1 Vectors and the Three Dimensional Space
8/27
UTP/JBJ 8
Theorem: An equation of the form
where E, F, Gand Hare constant,represents a sphere , a point , or has nograph.
0222 HGzFyExzyx
http://localhost/var/www/apps/conversion/tmp/scratch_10/C1L1%20Vectors%20and%20the%20Three%20Dimensional%20Space%20-%20Theorem.mwshttp://localhost/var/www/apps/conversion/tmp/scratch_10/C1L1%20Vectors%20and%20the%20Three%20Dimensional%20Space%20-%20Theorem.mws8/11/2019 C1L1 Vectors and the Three Dimensional Space
9/27
UTP/JBJ 9
Example 1
Find the center and radius of thesphere that has (1, -2, 4) and
( 3, 4, -12) as endpoints of adiameter.
Answers: C (h,k,l) =( 2, 1, -4)d =17.3 ; r = 8.6
8/11/2019 C1L1 Vectors and the Three Dimensional Space
10/27
UTP/JBJ 10
Example 2
Show that A(4, 5,2),B(1,7,3)and C(2,4,5) are vertices of
an equilateral triangle.
AB=AC=BC= 14
Answer:
8/11/2019 C1L1 Vectors and the Three Dimensional Space
11/27
UTP/JBJ 11
Simple Recall1. Find the distance between the points
A ( 2, 1, 3) and B (-1,4,1).
2. Find the equation of the sphere withcenter at the origin and passing through apoint (-1,-1, 2).
3. Find the center and radius of the sphere
given
11246222 zyxzyx
8/11/2019 C1L1 Vectors and the Three Dimensional Space
12/27
UTP/JBJ 12
VECTORS Definition
Vector is a quantity that has bothmagnitude and direction usually
represented by an arrow.
It has an initial point and a terminal point.
Vector quantity: velocity, displacement,force
8/11/2019 C1L1 Vectors and the Three Dimensional Space
13/27
UTP/JBJ 13
How to Draw a Vector? Example 1: v= in 3-
dimensional
Example 2: v= in 2-dimensional
Example 3 : a= 2i - 3j+ 4k where i,j,
kare unit vectors
8/11/2019 C1L1 Vectors and the Three Dimensional Space
14/27
UTP/JBJ 14
Equivalent Vectors Two vectors are equivalent if theircorresponding components are equal.
Example:
A= and B= < 2, -1, 3>
Vector A is equivalent to vector B !
8/11/2019 C1L1 Vectors and the Three Dimensional Space
15/27
UTP/JBJ 15
Sum of VectorsIf v and w are vectors , then the
sum v + w is
v+w
v
wv+w = w+v
0+v= v+0 =v
8/11/2019 C1L1 Vectors and the Three Dimensional Space
16/27
UTP/JBJ 16
Arithmetic Operations on
Vectors Theorem. If
spaceinwvwvwvwv
wwwwandvvvv
spaceforwvwvwv
wwwandvvv
3,,
,,,,
;2,
,,
332211
321321
2211
2121
8/11/2019 C1L1 Vectors and the Three Dimensional Space
17/27
UTP/JBJ 17
If v is a nonzero vector and k is anonzero real number (scalar) then thescalar multiple kv is defined to be avector whose length is k times thelength of v and whose direction is the
same as that of v if k > 0 and oppositethat of v if k< 0.
8/11/2019 C1L1 Vectors and the Three Dimensional Space
18/27
UTP/JBJ 18
spaceinwvwvwvwv
spaceinwvwvwv
3,,
2,
332211
2211
If k is any scalar, then
32121 ,,, kvkvkvkvorkvkvkv
8/11/2019 C1L1 Vectors and the Three Dimensional Space
19/27
UTP/JBJ 19
Vectors with Initial Point Not
the Origin Theorem.
space3in,,,ALSO
space2in,
),(pointterminaland),(
pointinitialwithspace2invectoraisIf
12121221
121221
222111
21
zzyyxxPP
yyxxPP
yxPyxP
PP
lengththeis|| 21PP
8/11/2019 C1L1 Vectors and the Three Dimensional Space
20/27
UTP/JBJ 20
Norm of a Vector Thenorm or the magnitude (length) is
denoted as
321
2
3
2
2
2
1
21
2
2
2
1
,,;space3in
,;space2in
vvvvvvvv
vvvvvv
8/11/2019 C1L1 Vectors and the Three Dimensional Space
21/27
UTP/JBJ 21
Example 3:
Find the norms of v=, 2v
and w=.
Answers:
58.4;66.112;83.5 wvv
8/11/2019 C1L1 Vectors and the Three Dimensional Space
22/27
UTP/JBJ 22
Unit Vectors In 2-space:
i = ; j =
In 3-space:i=; j= ; k=
i , j, k is known as the standardbasis vector
Example: A= = -i + 5j- 6z
8/11/2019 C1L1 Vectors and the Three Dimensional Space
23/27
UTP/JBJ 23
Normalizing a Vectorvkv
vu
1
is a unit vector with the same direction as v.
Example 4.Find the unit vector that has the samedirection as v=2i-j+4k
Answer: 4,1,221
1u
8/11/2019 C1L1 Vectors and the Three Dimensional Space
24/27
UTP/JBJ 24
Vectors Determined by
Anglev
y
x
cosv
sinv
jvivv
vv
sincos
sin,cos
v
8/11/2019 C1L1 Vectors and the Three Dimensional Space
25/27
8/11/2019 C1L1 Vectors and the Three Dimensional Space
26/27
8/11/2019 C1L1 Vectors and the Three Dimensional Space
27/27
UTP/JBJ 27
Sine Law
sinsinsin
cba
cos2222 abbac
Cosine Law
c
a
b