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- 6주차 강의 내용 -
Chap. 9 Vector Differential Calculus. Grad, Div, Curl
9.1 Vectors in 2-Space and 3-Space
There are two kinds of quantities in physics and geometry: scalars and vectors.
- Scalar ; magnitude ex) length, voltage, temperature, speed - Vector ; magnitude and direction
* arrow, directed line segment ex) force, velocity
Vector is denoted by lowercase boldface letters a, b, v, etc.
or
The magnitude (or length) of the vector a is denoted by Another name for length is norm (or Euclidean norm)
A vector of length 1 is called a unit vector.
a ,
ba .
Force and velocity
Definition
Equality of vectors
Two vectors a and b are equal, written a=b, if they have the same length and the same direction.
Components of a Vector
In an xyz Cartesian coordinate system in space, Let a be a given vector with initial point P : (x1, y1, z1) and terminal point Q : (x2, y2, z2). Then the three coordinate differences
(1) a1 = x2 - x1, a2 = y2 - y1, a3 = z2 - z1
Equal vectors a = b
(A)
Vectors having the same length but different directions
(B)
Vectors having the same direction but
different length (C)
Vectors having different length and
different direction (D)
(A) Equal vectors. (B)-(D) Different vectors
Chap. 9 Vector Differential Calculus. Grad, Div, Curl
The lengtha is,
(2) 32
2 2 2
1 a a
a
a
The position vector r of a point A : (x, y, z) is the vector with the origin (0, 0, 0) as the initial point and A as the terminal point. Thus r = [x, y, z]
Components of a vector Position vector r of a point A : (x,y,z)
Ex. 1) Find the unit vector a=[2,2,-1] and b=[5,-3,2]
3 )
1 ( 2
22 2 2
a b 52 (3)2 22 38
Chap. 9 Vector Differential Calculus. Grad, Div, Curl
3
, 1 3 , 2 3 2
ua
38
, 2 38 , 3 38 5 ub
Vector Addition, Scalar Multiplication
Addition of Vector
The sum a+b of two vectors a = [a1, a2, a3] and b = [b1, b2, b3] is obtained by adding the corresponding components,
(3) a+b = [a1+b1, a2+b2, a3+b3]
Geometrically, place the initial point of b at the terminal point of a ; then a+b is the vector drawn from the initial point of a to the terminal point of b.
For forces, this addition follows parallelogram law by which we obtain the resultant of two forces in mechanics.
Geometrical meaning of vector addition Resultant of two forces (parallelogram law)
Chap. 9 Vector Differential Calculus. Grad, Div, Curl
Basic Properties of Vector Addition
(4)
(a) a + b = b + a (Commutative)
(b) (u + v) + w = u + (v + w) (Associative) (c) a + 0 = 0 + a = a
(d) a + (-a) = 0
Here -a denotes the vector having the length and the direction
opposite to that of a. a
Definition
Scalar Multiplication (Multiplication by a Number)
The product ca of any vector a = [a1, a2, a3] and any scalar c (real number c) is the vector obtained by multiplying each component of a by c,
(5) ca = [ca1, ca2, ca3]
Geometrically, if a≠0, then ca with c>0 has the direction of a and with c<0 the direction opposite to a. In any case, the length of ca is
, and ca=0 if a = 0 or c = 0 (or both).
a c ca
Chap. 9 Vector Differential Calculus. Grad, Div, Curl
Basic Properties of Scalar Multiplication
(6)
(a) c(a + b) = ca + cb
(b) (c + k)a = ca + ka
(c) c(ka) = (ck)a ( written cka )
(d) 1a = a
(7) (a) 0a = 0
(b) (-1)a = -a
Unit Vectors i, j, k
Besides a = [a1, a2, a3], another way of writing vectors is (8) a = a1i + a2j + a3k
i, j, k are unit vector in the positive directions of the axes of a Cartesian coordinate system.
(9) i = [1, 0, 0] j = [0, 1, 0] k = [0, 0, 1] The unit vector i, j, k and the representation
Chap. 9 Vector Differential Calculus. Grad, Div, Curl
Chap. 9 Vector Differential Calculus. Grad, Div, Curl
Ex. 2) If a=[4,0,3] and b=[-2,1,5], Find a+b, 2a-3b a+b = [4+(-2),0+1,3+5] = [2,1,8]
2a-3b = 2[4,0,3]-3[-2,1,5] = [8,0,6]-[-6,3,15] = [14,-3,-9]
Ex. 3) If a=i+2j-3k and b=4i+7k, Find 2a+3b 2a+3b = 2(i+2j-3k)+3(4i+7k)= 14i+4j+15k
Ex. 4) Find the unit vector in the direction of the vector a=2i-j-2k 3
) 2 ( ) 1 (
22 2 2 a
The unit vector in the same direction is k
j i k j i ua
3 2 3 1 3 ) 2 2 2
3(
1
9.2 Inner Product (Dot Product)
Definition
Inner Product (Dot Product) of Vectors
The inner product or dot product a·b of two vectors a and b is the product of their lengths times the cosine of their angle.
(1) a·b = cos if a≠0, b≠0
a·b = 0 if a=0, or b=0
The angle , 0≤ ≤π , is between a and b. In components, (2)
b
a
a1,a2,a3
, b
b1, b2,b3
,a aba1b1 a2b2 a3b3
A vector a is called orthogonal to a vector b if a·b = 0. ← Orthogonal vectors
Angle between vectors and value of inner product
Chap. 9 Vector Differential Calculus. Grad, Div, Curl
Length and Angle
(3) a aa
(4) a a b b
b a b
a b a
cos
For any vectors a, b, c and scalars q1, q2,
(5)
(a) (Linearity)
(b) (Symmetry)
(c) (Positive-
definiteness) if and only if
c b q c a q c a q a
q1 2 ) 1 2 (
a b b a
0
a a
0
a
a a0
Theorem 1
Orthogonality
The inner product of two vectors is 0 if and only if these vectors are perpendicular.
Here dot multiplication is commutative and is distributive with respect to vector addition. From (5a) withq1 1 and q2 1, we have
(5a*) (a )b cacbc (Distributive)
Chap. 9 Vector Differential Calculus. Grad, Div, Curl
From (1) and cos 1, (6)
(7) (8)
b a b
a
b a
b
a
) (
2 2 2
2
2 a b a b
b
a
(Cauchy-Schwarz Inequality) (Triangle Inequality)
(Parallelogram equality)
Ex. 1) Find the angle between a=[2,2,-1] and b=[5,-3,2]
b b a a
b a b
a b a
cos3 ) 1 ( 2
22 2 2
a b 52 (3)2 22 38
2 2 ) 1 ( ) 3 ( 2 5
2
a b
38 3
cos γ 2 ) 84
38 3 ( 2 cos 1 γ
Ex. 2) Show that a=[2,2,-1] is perpendicular to b=[5,-4,2]
0 2 ) 1 ( ) 4 ( 2 5
2
b a
Chap. 9 Vector Differential Calculus. Grad, Div, Curl
9.3 Vector Product (Cross Product)
Definition
Vector Product (Cross Product, Outer Product) of Vectors
The vector product (also called cross product or outer product) a×b of two vectors a and b is the vector v = a × b.
If a and b have the same or opposite direction, or if a=0 or b=0, then v= a×b = 0.
In any other case v= a×b has the length (1)
This is the area of the parallelogram. The direction of v= a×b is perpendicular to both a and b and such that a, b, v in this order, form a right-handed triple or right-handed screw.
γ b
a b a
v sin
Let a = [a1, a2, a3] and b = [b1, b2, b3]. Then v = [v1, v2, v3] = a×b has the components
(2)
v
1= a
2b
3- a
3b
2, v
2= a
3b
1- a
1b
3, v
3= a
1b
2- a
2b
1Chap. 9 Vector Differential Calculus. Grad, Div, Curl
Right-Handed Cartesian Coordinate System
The system is called right-handed if the corresponding unit vectors i, j, k in the positive direction of the axes form a right-handed triple.
The system is called left-handed if the sense of k is reversed.
We prefer right-handed system.
in (2), (2*)
(2**)
2 1
2 1 3 1 3
1 3 3
1 3 1 2
3 2
3 2
1 , ,
b b
a v a
b b
a a b
b a v a
b b
a
v a
2 1
2 1 3
1 3 1 3
2 3 2
3 2 1
3 2
1 b b
a ka b b
a ja b b
a ia b b b
a a a
k j i b a
v
Ex. 1) Find a×b, when a=[1,3,4] and b=[2,7,-5]
k j i k
j i
k j i b
a
43 13
7 2
3 1 5 2
4 1 5 7
4 3 5 7 2
4 3 1
Vector product
Right-handed triple of a vectors a, b, v
Right-handed screw
Chap. 9 Vector Differential Calculus. Grad, Div, Curl
THEOREM 1
General Properties of Vector Products (a)
For every scalar ,
(4) ( a) × b = (a×b) = a × ( b)
(b)
Cross multiplication is distributive with respect to vector addition.
(5)
(c)
Cross multiplication is not commutative but anti- commutative.
(6) b × a = - (a × b)
(d)
Cross multiplication is not associative
(7) a × (b × c ) ≠ (a × b) × c) (
) (
) (
)
( α a b c a b a c ) (
) (
) (
)
( β a b c a c b c
(3) i×j = k, j×k = i, k×i = j j×i = -k, k×j = -i, i×k = -j
Anticommutative of cross multiplication
l
l l l
a×b
a b
Chap. 9 Vector Differential Calculus. Grad, Div, Curl
Properties and Applications of Scalar Triple Products (a)
In (10) the dot and cross can be interchanged :
(11) (a b c) = a·(b×c) = (a× b)·c(b) Geometric interpretation
The absolute value
l(a b c)lof (10) is the volume of the parallelepiped (oblique box) with
a, b, cas edge vectors
.Scalar Triple Product
The scalar triple product or mixed triple product of a, b, c is denoted by (10*) (a b c) = a·(b×c) ← scalar
a·(b×c)= a·v = a1v1a2v2 a3v3← from (2) in 9.2 and (2*) in 9.3
3 2 1
3 2 1
3 2 1
2 1
2 1 3 1 3
1 3 2 3 2
3 2 1
c c c
b b b
a a a c c
b a b c c
b a b c c
b
a b
(10)
3 2 1
3 2 1
3 2 1
) ( ) (
c c c
b b b
a a a c b a c b
a
THEOREM 2
Geometrical interpretation of a scalar triple product
Chap. 9 Vector Differential Calculus. Grad, Div, Curl
Ex. 2) Find a vector perpendicular to the plane that passes through the points P(1,4,6), Q(-2,5,-1), and R(1,-1,1)
k j i k
j
i 40 15 15
5 0
1 3 5
0 7 3 5
5 7
1
Chap. 9 Vector Differential Calculus. Grad, Div, Curl
The vector PQPR is perpendicular to both and PQ PR and is therefore perpendicular to the plane through P, Q, and R.
k j i
PQ3 7 PR5j5k
5 5 0
7 1 3
k j i PR PQ
Ex. 3) Find the area of triangle with vertices P(1,4,6), Q(-2,5,-1), and R(1,-1,1) In Ex. 2) PQPR40i15j15k.
The area of parallelogram with adjacent sides PQ and PR is the length of this cross product, PQ PR (40)2(15)2152 5 82
Thus, the area of triangle is 2.5 82
Chap. 9 Vector Differential Calculus. Grad, Div, Curl
Ex. 3) Use the scalar triple product to show the vectors a=[1,4,-7], b=[2,-1,4]
and c=[0,-9,18] are coplanar.
18 9 0
4 1 2
7 4 1 ) ( ) (
a b c c
b a
0 ) 18 ( 7 ) 36 ( 4 9 18
0 1 72
18 0
4 42 18 9
4
1
The volume of the parallelepiped determined by a, b, c is 0.
This means that the vectors a, b and c are coplanar.
