#09 (09/20/2023)

Vector Analysis

Scalar product (inner product, dot product)

The scalar product between the two vectors, a and b, is defined as

a·b = a b cos θ,
(1)
where θ is the angle between the two vectors.
innerpro.jpg
The following properties can be derived based on the definition of Eq.(1).
  1. a ·b = b ·a
  2. c a ·b = a ·c b = c (a ·b)
  3. a ·(b + c ) = a ·b + a ·c
It is possible to derive the scalar product in terms of the components of a vector by introducing the base vectors, ex and ey. Note that

ex·ex
=
ey·ey
=
1,
ex·ey
=
ey·ex
=
0.
(2)
Using Eq.(2), the scalar product of two vectors in components is
a·b
=
(ax ex +ay ey )·(bx ex +by ey )
=
ax bx ex ·ex + ax by ex ·ey + ay bx ey ·ex + ay by ey ·ey
=
ax bx + ay by.
(3)
The definition of Eq.(1) is a preferred form as it is independent of the coordinate system.

Vector product (cross product)

The vector product is defined between two vectors in 3-D as

a×b ≡ S n,
(4)
where S is the area of the parallelogram spanned by the two vectors and n is the unit vector whose direction is in the right hand system as shown in the figure.
crosspro.jpg
The following properties can be derived based on the definition of Eq.(4).
  1. a ×b = −b ×a
  2. c a ×b = a ×c b = c (a ×b)
  3. a ×(b + c ) = a ×b + a ×c
Note that a×a = 0 (no area by two identical vectors).
The relationship among 3-D base vectors is summarized as

ex×ey
=
−ey×ex
=
ez
ey×ez
=
−ez×ey
=
ex
ez×ex
=
−ex×ez
=
ey,
(5)
and

ex ×ex = ey ×ey = ez ×ez = 0.
(6)
Using the relationship above, the vector product is expressed in components as
a×b
=
(ax ex +ay ey +az ez )×(bx ex +by ey +bz ez )
=
ax bx ex×ex+ax by ex×ey+ax bz ex×ez+ …
=
…
=
⎢
⎢
⎢
⎢
⎢
ex
ey
ez
ax
ay
az
bx
by
bz
⎢
⎢
⎢
⎢
⎢ 		.
(7)

Equation of plane (application of dot product)

eqplane.jpg
The relationship among x0 (fixed point), x (moving point) and n (normal, fixed vector) is

n ·(x−x0) = 0.
(8)
So the equation of a plane is expressed as
n ·x
=
n ·x0
=
c.
(9)
Note that c represents the shortest distance between the plane and the origin.

Triple products

There are two types of vector products, scalar triple product and vector triple product, whose main usage is found in vector analysis.

Scalar triple products, a·(b×c)

a·(b×c) = a·S n = S a cosθ = S h represents the volume of the parallelepiped spanned by the three vectors, a, b and c. Note the cyclic relationship below.
a·(b×c) = b·(c×a) = c·(a×b)
triplepro.jpg

Vector triple products, a ×(b×c)


a ×(b×c) = (a·c)b − (a·b)c.
Note that the right hand side is a combination of b and c both of which are inside the parentheses in the left hand side. The one that is in the middle (b) comes first.
Exercise
  1. a×( b×c) +b×( c×a) +c×( a×b) = ?
  2. (a×b)·(c×d) = ?


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On 25 Sep 2023, 23:11.