Cal
#
, Ch12 : Vectors a
Geometry of Space
12 . 1) 3-D Coordintes ConstantFunction x +
y = r?
#
2n
↓ (3 ,
(x y z)
1
, 1) (0 ,
(X y z)
-1
,
-
1)
y
, , , ,
y =
4
3.D :
Distance in D= /
4 -x ) (yz y i + (22 2 72 ,
+ -
,
-
, (1 ,
2
,
3) (5 , 7 9)
,
↳/
-
5 + 17*2) +(973) 46
+ 25536)
2
-
>
-
T
Equation
of Sphere : (x-h) <
ly-m) +(2-1)" = r2Sphereer (h k , .
1) with radius (2)
-
(2) -
> 22 - 4
-
- 32 - + 9
↑
12 > /
-
# x2 y +Em + #x- 6 -22 +
-m
+ 6 = 0 - (x (4x) +
(y Gy) =
+ (2) + 60 6
22) -
-
men
(x 4x 4) (yz by 9) + (2) 22 1) (X + 2) 3) + (2 12 ( 1) 58
+ + + + + + 6 > +
(y 8 - 2 3
- = - -
+ =
- -
=
, ,
~
formula
(x + y
=
=
4)8(y + 22 =
16) -
prefect clynder (y -y 16) 8(y2 = x) >
- Cylinder
right
= but still
12 2) Vectors
.
↳
(4 5 6)
Vectors : A w/
quanity magnitude and direction.
, ,
a a
V = (4 +
1
,
5 -
2
,
6 -
3) >
-
43 ,
3
,
3) * (1, 2 ,
3)
aT
Zero Vector 10 ,
0, 0 +
a
B
&
> D
Vector Addition : Component Wise (1 ,
2
,
37 + (6 ,
7
,
4) >
-
(156 ,
252, 3 + 4) = (7, 9
,
7)
Scalar Multiplication : ((a .,
02 , 03) -
> <29. ,
Ca2 ,
Cas) 2( ,
0 ,
1) = <- 2 ,
0
,
2)
Length : =
(a , Az 93] , Standard Basis Vector:
ll =a+A (1 2 , 3)
,
= 1 +
2y + 3k
A of of 1
unit vector is a
If t is a nonzero
vector
Length = /11
.
(iv)t has of 1
vector, then vector
length
=
the .
All
V
has the same direction as .
Proof. Since i 0 and I has S same direction as V .
1 lull Ill Illi 1(11 i xilvll 1
= = x = =
To find the unit vector direction V divide i it's
magnitude
in same as ,
by .
Find a unit vector in direction of V =
<2 5) ? ,
i | ill = N+ 5 + M
+2 +9 =( -
2
,
5) - -Yea ,
5/2a]
Find a unit vector in direction of V <3 5 47 ? =
25 25 ,25s]
, ,
5+ 4 - T
11 ill 15 =
+ 25T6 50 Es 1 - 55(3 + + +
,
5, 4) -
,
, 12 . 3) Dot Product
(b babs
If
if <a. ,
an ,
ad +
,
are vectors
a .
b = a, b , + a zbz + AzDz
Ex1)(1 ,
2) (1 5) ,
+ -
1 + 10 + 9 Ex2)(1 ,
0, 07 (0 ,
1
,
0 =3 0
Properties of Dot Producti -a .
a =
1ak -
a .
b = b .
a -
a .
(bt) = a .
b + a .
c
-
Cla.b) =
((a)b =
a .
ch -
a .
8 = 0
Thmi If E is the
angle between att ,
than a . b = lallbICost
/Fact : The two vectors are
orthogonal It angle between them is TY2]
Two vectors are orthogonal if Gob =
Ex 3) 2ec2y- Se
4g 21 >
-
+ -
101 -
82- 2k -
> 0
Orthogonal
Ex4)(2 ,
2
,
-1) (5 ,
-3
,
2) Ia+ 3 I bla 58 + Cost =
a
Yallbl
Cost = 10 -
6 -2/3538 + 2/358 - Cost (73558) => 1 46.
Direction Angles : The direction
angle is the is
angle between the vectors of X,
y ,
z axis ·
2 =
au/a)
Ex5) <1 , 2 , 3) cost =
Ya-Yi CosB=/Cost 3/4 =
"
Projectioni , in Scalar ProjiCompat =
a
-al Vector Proje Projab = 19 .ak) a
Ex6)b <1 1 2) a= < 2 3 1) Kal = + Compan = >
3/4
-
= -
, , , ,
(3/14) ( 1) 2 6/4 , 94 3/4)
Projob 2 3
-
=
-
+
, ,
,
12 . 4) Cross Product Determinants
cXd
a-ba
3x3Ce(h-ye) Ex1)(23 1(45x8) ael-ahf-bdi
A
bhf 4
+
-
3 + 12
3
-
9
-
2(352) + 313273
&
6 -
3
+ +
can-eye >
- - -
(Fact : axb is
orthogonal to both a + b)
3
iik 11 6) + k( 3)
Cross Product angle Ex2)
-
& If t axb = e(5(8)
Trig :
>
the
- - - -
is
01 2
between b
at then laxbl = lallbISinG 345 -
32 +
61 -
3k
#: add are parallel if their cross product is zero . )
97
axb =
area of
parallogram determined
by ath 3
means perpendicar
-
Ex3) Find 4 , 6) (-2 -5, -1), and (1 -1 , 1)
vector the plane Containing (1
a to
, , , ,
12k
146
>
-
2(4 + 6) y(1 6) + k( 4) -
- -
(Divide
[multplyYz)
2) or
zuz
1 -
11
↳
102 + 51-523400r2sres Properties of Cross Product
its a trangle so (12)
i >
-
Est > 55(1) -
= (55)0axb = -
bxa ⑨ a =
(b xc) =
(axb) ·
C
② Caxb = axcb =
c(axb)Gax(bxc) = (a c)b . -
(a .
b)c
③ ax(b +) = axc + axb
Fact : Thecalar
triple product) gives the volume of parallel b, C
the piped determined by a ,
.
Fact : If Sip zero rectors are co-planar
is .
I
Ex4)(1 ,
4 , 7) (2 , ,
-
1
,
4) 40 ,
,
-9 ,
18)
14 -7
2 + 4
+ 17 %
36) -
4(36)
↑
-
7 ( 18)
12 - 144-144
0-918 18-144 +
#
, Ch12 : Vectors a
Geometry of Space
12 . 1) 3-D Coordintes ConstantFunction x +
y = r?
#
2n
↓ (3 ,
(x y z)
1
, 1) (0 ,
(X y z)
-1
,
-
1)
y
, , , ,
y =
4
3.D :
Distance in D= /
4 -x ) (yz y i + (22 2 72 ,
+ -
,
-
, (1 ,
2
,
3) (5 , 7 9)
,
↳/
-
5 + 17*2) +(973) 46
+ 25536)
2
-
>
-
T
Equation
of Sphere : (x-h) <
ly-m) +(2-1)" = r2Sphereer (h k , .
1) with radius (2)
-
(2) -
> 22 - 4
-
- 32 - + 9
↑
12 > /
-
# x2 y +Em + #x- 6 -22 +
-m
+ 6 = 0 - (x (4x) +
(y Gy) =
+ (2) + 60 6
22) -
-
men
(x 4x 4) (yz by 9) + (2) 22 1) (X + 2) 3) + (2 12 ( 1) 58
+ + + + + + 6 > +
(y 8 - 2 3
- = - -
+ =
- -
=
, ,
~
formula
(x + y
=
=
4)8(y + 22 =
16) -
prefect clynder (y -y 16) 8(y2 = x) >
- Cylinder
right
= but still
12 2) Vectors
.
↳
(4 5 6)
Vectors : A w/
quanity magnitude and direction.
, ,
a a
V = (4 +
1
,
5 -
2
,
6 -
3) >
-
43 ,
3
,
3) * (1, 2 ,
3)
aT
Zero Vector 10 ,
0, 0 +
a
B
&
> D
Vector Addition : Component Wise (1 ,
2
,
37 + (6 ,
7
,
4) >
-
(156 ,
252, 3 + 4) = (7, 9
,
7)
Scalar Multiplication : ((a .,
02 , 03) -
> <29. ,
Ca2 ,
Cas) 2( ,
0 ,
1) = <- 2 ,
0
,
2)
Length : =
(a , Az 93] , Standard Basis Vector:
ll =a+A (1 2 , 3)
,
= 1 +
2y + 3k
A of of 1
unit vector is a
If t is a nonzero
vector
Length = /11
.
(iv)t has of 1
vector, then vector
length
=
the .
All
V
has the same direction as .
Proof. Since i 0 and I has S same direction as V .
1 lull Ill Illi 1(11 i xilvll 1
= = x = =
To find the unit vector direction V divide i it's
magnitude
in same as ,
by .
Find a unit vector in direction of V =
<2 5) ? ,
i | ill = N+ 5 + M
+2 +9 =( -
2
,
5) - -Yea ,
5/2a]
Find a unit vector in direction of V <3 5 47 ? =
25 25 ,25s]
, ,
5+ 4 - T
11 ill 15 =
+ 25T6 50 Es 1 - 55(3 + + +
,
5, 4) -
,
, 12 . 3) Dot Product
(b babs
If
if <a. ,
an ,
ad +
,
are vectors
a .
b = a, b , + a zbz + AzDz
Ex1)(1 ,
2) (1 5) ,
+ -
1 + 10 + 9 Ex2)(1 ,
0, 07 (0 ,
1
,
0 =3 0
Properties of Dot Producti -a .
a =
1ak -
a .
b = b .
a -
a .
(bt) = a .
b + a .
c
-
Cla.b) =
((a)b =
a .
ch -
a .
8 = 0
Thmi If E is the
angle between att ,
than a . b = lallbICost
/Fact : The two vectors are
orthogonal It angle between them is TY2]
Two vectors are orthogonal if Gob =
Ex 3) 2ec2y- Se
4g 21 >
-
+ -
101 -
82- 2k -
> 0
Orthogonal
Ex4)(2 ,
2
,
-1) (5 ,
-3
,
2) Ia+ 3 I bla 58 + Cost =
a
Yallbl
Cost = 10 -
6 -2/3538 + 2/358 - Cost (73558) => 1 46.
Direction Angles : The direction
angle is the is
angle between the vectors of X,
y ,
z axis ·
2 =
au/a)
Ex5) <1 , 2 , 3) cost =
Ya-Yi CosB=/Cost 3/4 =
"
Projectioni , in Scalar ProjiCompat =
a
-al Vector Proje Projab = 19 .ak) a
Ex6)b <1 1 2) a= < 2 3 1) Kal = + Compan = >
3/4
-
= -
, , , ,
(3/14) ( 1) 2 6/4 , 94 3/4)
Projob 2 3
-
=
-
+
, ,
,
12 . 4) Cross Product Determinants
cXd
a-ba
3x3Ce(h-ye) Ex1)(23 1(45x8) ael-ahf-bdi
A
bhf 4
+
-
3 + 12
3
-
9
-
2(352) + 313273
&
6 -
3
+ +
can-eye >
- - -
(Fact : axb is
orthogonal to both a + b)
3
iik 11 6) + k( 3)
Cross Product angle Ex2)
-
& If t axb = e(5(8)
Trig :
>
the
- - - -
is
01 2
between b
at then laxbl = lallbISinG 345 -
32 +
61 -
3k
#: add are parallel if their cross product is zero . )
97
axb =
area of
parallogram determined
by ath 3
means perpendicar
-
Ex3) Find 4 , 6) (-2 -5, -1), and (1 -1 , 1)
vector the plane Containing (1
a to
, , , ,
12k
146
>
-
2(4 + 6) y(1 6) + k( 4) -
- -
(Divide
[multplyYz)
2) or
zuz
1 -
11
↳
102 + 51-523400r2sres Properties of Cross Product
its a trangle so (12)
i >
-
Est > 55(1) -
= (55)0axb = -
bxa ⑨ a =
(b xc) =
(axb) ·
C
② Caxb = axcb =
c(axb)Gax(bxc) = (a c)b . -
(a .
b)c
③ ax(b +) = axc + axb
Fact : Thecalar
triple product) gives the volume of parallel b, C
the piped determined by a ,
.
Fact : If Sip zero rectors are co-planar
is .
I
Ex4)(1 ,
4 , 7) (2 , ,
-
1
,
4) 40 ,
,
-9 ,
18)
14 -7
2 + 4
+ 17 %
36) -
4(36)
↑
-
7 ( 18)
12 - 144-144
0-918 18-144 +
