Multivariable Differential Calculus for Engineering

, &




Q101/16
20 min /I problems
Content :

3 dim coord system (distance , Sphere
Vectors
Dot products

3 dim coordinate
system




I
Geometry Equation ,
coordinate ,
algebra
Point S
- (a b c)
, ,
on R3 (9 , b) on R2


Schere c) (y b) (z c)
2
- (x -
+ -
+ -
= ru
Center : (a, b , c)
Radius : r



E g
.


.
S (-1) y2
: + + z = 1 smallest distance A and
A 13 0 0
:
, ,
a point on Sphere S



i J

↑
(3 0 0 find distance between

S
,

center of S and A
,




↳
subtract radius

,Vectors :
directed line segment AB
terminal &


-B
point
"D
Magnitude (length) AB of
=
/A/
A
S
C
·
Two vectors AB & D
initial
point
are equal if they have same length and
direction

"Vector the standard position
"
-

in
initial point at origin
↳

-



"Component form"
i 0 = with 0 10 0 01 and lab, c) =

, ,
=



then the component form of < a b c > =
, ,



-

Eg
. .
#B =? component form ofis
↑ (1 2 3) B(2 3 4)
, , AB , ,
= <7 - 1 , 3-2 4-33
,

= [1 , 1 , 13


product
Not (Requires ICTORS] 7
i
= I/II COSE > & O-& T
i
-.
Mis
=
scalar NOT vector
distance
Why ?
& Properties : lik = T i
:




lulose i (i) =
T+ .

, The that
Projections
of in ontois parala
:
projection a vector is



B
F ProjB Projec
-

U = =




ii
F
3
D



component of i

IProjc) =
iCOSE Since Project has same direction
scar as i

Proje =
-Vector

Inos
=




= (i) =
More properties of dot product :




① (1 =
3 (l =
2 ② Dot product o
Can = 7 ? NO = I
= III cose
..

-
III