Calculus III: Multivariable Calculus

, Ch 12 2 .
.
-
Vectors , Vector Addition

Magnitude/Normalization
V= [V V2 V3]
, , ,




DEF : (vl =
/V ,
2
+ v22 V+




↳ Direction :
R =
tan" (2)
Vector Operators
1.) Vector Addition U= ( , , 42) v= (V V2]
,,


FORMULA :
y+v = [u ,
+ V
,, z + Vz)
↳ i e . .
tip to tail method
eu =
<3 , 4) v= <1 2) ,




U +v = (3 +1 ,
4 + 2) =
44 , 6)
2) Scalar Multiplication
V= (vi ,2)
FORMULA : for Scalar <Erector
(v =
[cv ,
+
cVz]
c= 3 <1 2)
e v= ,



3V =
73 .
1
,
3 .

27 =
(3 67 ,

, Ch 12 3 Dot Product
. .
-




The Dot Product

DEF : =
(n , ,
427 =
<V N2) , ,




Y Y . =
1 , V, +
12V

Geometric
-
Interpretation
8
↳ the dot product relates to the angle
between the vectors :


. = /cos8
* F i. = 0 than the vectors are


: u =
(2 3) ,
Y 14 17
=
,




u 8 .
=
2 4 . + 3 1 . = 1
me




i =z
cost +
=
21
2 (2 37 41 07
=
,
·

,

, Vector Projection
DEF : the projection of rector a onto v :

U
u v
↳ Projec

4jection
.

=



lar
e i (3 =
,
47 =
(1 27
,

7
-

u V=.
3+ 8 = 11

Iv/2 = 12 2 +
2
=
5
projru = (1 2) =
,




Scalar Projection
* AKA component
DEF : the component of a
along v:



↳ H V
.




comput =


iv.