1. Scalar and vector quantities
- Scalar : A physical quantity specified by a single magnitude (not by
a direction). For examples, density, volume, temperature, length, and so on.
- Vector : A physical quantity expressed by a magnitude and a
direction. For examples, position, displacement, velocity, acceleration, momentum,
and force.
2. Vector notations
- Vector quantities are denoted by A (bold) or
 (arrow).
- A vector can be specified by its components which are different depending on a
particular coordinate system. For example, a position vector A
is described by [Ax, Ay, Az] in 3D-Cartesian Coordinate System.
3. Definitions and rules
- Equality : Vector A and vector B
are equal if, and only if, their representative components are the same. In other words A
= B means [Ax, Ay, Az] = [Bx, By, Bz], which is equivalent to Ax = Bx, Ay
= By, and Az = Bz.
- Addition : The addition of two vectors is a vector whose components
are sums of the components of the vectors. Therefore
- Multiplication by a scalar : cA = c[Ax, Ay, Az] =
[cAx, cAy, cAz] = Ac.
4. Scalar product (dot product)
- Definition :
A · B = AxBx + AyBy + AzBz
(5.1)
- Rules :
A · B
= B · A
(5.2)
A ·(B + C)
= A · B +
A · C
(5.3)
A2
= |A|=
A · A
(5.4)
- Cosine of a angle bet. two lines (another definition of dot product)
:
A · B
= |A||B|cos q
(5.5)
- Geometrical explanation : A · B
is the projection of A on B times B.
So if A is perpendicular to B, A · B is zero.
- Projection : One important use of dot products is in projections. The
scalar projection of b onto a is the length of the segment AB
shown in the figure below. The vector projection of b onto a is the
vector with the length that begins at the point A points in the same direction (or
opposite direction if the scalar projection is negative) as a.
Fig.5.1
Thus, mathematically, the scalar projection of b onto a is |b|cos q which is given by
(5.6)
This quantity is also called the component of b in the a direction (hence
the notation comp). And, the vector projection is merely the unit vector a/|a|
times the scalar projection of b onto a:
(5.7)
Thus, the scalar projection of b onto a is the magnitude of the vector
projection of b onto a.
- Conceps applied for unit vectors :
i · i
= j · j = k · k =
1
(5.8)
i · j
= i · k = j · k =
0
(5.9)
Where i, j, k are the unit vectors correponding each to x-axis, y-axis, and z-axis.
- Example :
(1) Work
Suppose you want to find the work W done in moving an object from one
point to another. As you know, W=Fd where F is the magnitude of the force moving the
object and d is the distance between the two points. However, this relation is only valid
when the force acts in the direction the particle moves. Suppose this is not the case. Let
the force vector be F=[2, 3, 4] and the displacement vector be d=[1, 2, 3].
In this case, the work is the product of the distance moved (the magnitude of the
displacement vector) and the magnitude of the component of the force that acts in the
direction of displacement (the scalar projection of F onto d):
(5.10)
Thus, the work done by the force to displace the object from the origin to
the point (1,2,3) is
As you see above, using a angle between the vectors F and d
is another way to compute the work.
(2) Law of cosines :
C = A +
B
C · C = (A + B) ·
(A + B) = A·A + 2AB +
B · B
C2 = A2
+ 2AB cos q + B2
Fig.5.1
5. Vector product (cross product)
There are two types of multiplication of vectors by other vectors. One type, the dot
product, is a scalar product which is shown above; the result of the dot product of two
vectors is a scalar. The other type, called the cross product, is a vector product, which
means it produces another vector rather than a scalar.
- Definition
The cross product of two vectors a=[a1, a2, a3 and b=[b1, b2, b3]
is given by
(6.1)
The easier way to remember the formular for the cross product is using a
determenant. Recall that the determinant of a 2 x 2 matrix is
(6.2)
and also the determinant of a 3 x 3 matrix is
(6.3)
Now we can write the formular for the cross product as below.
 (6.4)
- Example
The cross product of the vectors a= [3,-2,-2] and b= [-1,0,5] is
(6.5)
Properties of the Cross Product:
- The magnitude of the cross product of two vectors is
- The magnitude of the cross product of two vectors is equal to the area of the
parallelogram determined by the two vectors
- Anticommutativity is applied to the cross product
- Multiplication by scalars
- Distributivity
- The scalar triple product of the vectors a, b, and c
- The volume of the parallelepiped determined by the vectors a, b, and c
is the magnitude of their scalar triple product.
- The vector triple product of the vectors a, b, and c
Fig.6.1
Note that the result for the length of the cross product leads directly to the fact
that two vectors are parallel if and only if their cross product is the zero vector. This
is true since two vectors are parallel if and only if the angle between them is 0 degrees
(or 180 degrees).
- Example
(1) Area of a triangle
Suppose that a triangle has vertices (1,1,3), (4,-1,1), and (0,1,8) in three dimensions
and we want to find the area of the triangle. We can make two vectors a
and b which are a from (1,1,3) to (4,-1,1) and the vector b
from (1,1,3) to (0,1,8). Then a = [4-1, -1-1, 2-3] = [3,-2,-2] and b = [-1,
0, 5]. therefore the parallelogram determined by a and b is
Sincee the area of the triangle of if a half of that of the parallelogram, the area of
the traingle is 8.26.
(2) Torque
Fig.6.2
When a single force F is applied on
the point of P of an object which is free to turn about a fixed point O, the object tends
to rotate and the direction of rotation is along the axis of rotation. The magnitude of
the torque is given by
|N| = |rr x FF|= r F sinq
(6.6)
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