Chapter 1 Vectors
1- Vector and scalar quantities
A scalar quantity has only magnitude and no direction
A vector quantity has both magnitude and direction
The number of apples is an example of a scalar quantity. If you are told there are 38 apples in the basket, this completes the required information; no specification of direction is required. Other examples of scalar quantities are temperature, volume, mass, and time intervals.
Force is one example of a vector quantity. To describe completely the force on an object, we must specify both the direction of the applied force, a number to indicate the magnitude of the force. Velocity is another example of a vector quantity. If we wish to describe the velocity of a moving car, we must specify both its speed (say, 25m/s) and the direction in which the car is moving (say, southwest).
In this text, we use an arrow over the letter for vector notation. The magnitude of the vector is written A. The magnitude of a vector has physical units, such as meters for displacement or meters per second for velocity.
1-1- Some properties of vectors
- Equality of vectors
only if magnitude of is equal to magnitude of (A = B) and has the same direction as to .
Figure 1. These four vectors are all equal since they have equal lengths and point in the same direction.
- Addition
When vector is added to vector , the resultant is the vector that runs from the tail of to the tip of .
Figure 2
Figure 3
- Negative of a vector
Figure 4
(1-1)
The vectors and - have the same magnitude but point in opposite directions
- Substraction of vectors
Figure 5
(1-2)
- Multiplication of a vector by a scalar
Figure 6
If is multiplied by a positive scalar quantity m, the product m is a vector that has the same direction as and magnitude mA. If m is a negative scalar quantity, the vector m is directed opposite
1-2- Unit vector
A unit vector is a dimensionless vector having a magnitude of exactly one. Unit vectors are used to specify a given direction and have no other physical significance. They are used solely as a convenience in describing a direction in space. We shall use the symbols, and to represent unit vectors pointing in the positive x, and ydirections, respectively.
Figure 7. The unit vectors , and are directed along the x, and y axes, respectively
The unit vectors , and form a set of mutually perpendicular vectors in a right-handed coordinate system as shown in Figure 9. The magnitude of each unit vector equals unity; that is,
(1-3)
1-3- Components of a vector
Consider a vector lying in the xy plane and making an arbitrary angle with the positive x axis, as in Figure 10. The product of the component Ax and the unit vector is the vector Ax, which is parallel to the x axis and has magnitude Ax. Likewise, Ay is a vector of magnitude Ayparallel to the y axis. Thus, the unit-vector notation for the vector is written
(1-4)
Figure 8
Ax and Ay are the components of the vector .
Ax represents the projection of along the x axis and Ay represents the projection of along the y axis
Ax and Ay can be positive or negative.
(1-5)
(1-6)
(1-7)
(1-8)
Suppose we wish to add vector to vector , where has components Bx and By. The procedure for performing this sum via the component method is to simply add the x and y components separately. The resultant vector is therefore
(1-9)
Since , we see that the components of the resultant vector are
(1-10)
The magnitude of and the angle it makes with the x axis can then be obtained from its components using the relationships
(1-11)
and
(1-12)
2. Multiplying a vector by a vector
2.1- The scalar product
The scalar product (or dot product) of any two vectors and (read '' dot '') is a scalar quantity equal to the product of the magnitudes of the two vectors and the cosine of the angle between them:
(1-13)
where the result is a scalar quantity and is the angle between the directions of the two vectors.
The scalar product obeys the commutative law
(1-14)
and distributive law
(1-15)
The unit vectors and , lie in the positive x and y directions, respectively. Therefore, it follows from the definition of that the scalar products of these unit vectors are
(1-16)
Two vectors and can be expressed in component vector form as
(1-17)
Therefore, we can reduce the scalar product of and to
(1-18)
2.2 vector product
The vector product of two vectors and is a vector: ( vectorial )
The magnitude of this vector is (θ being the angle between and )
(1-19)
Vector products of the unit vectors:
In three dimensions, Any vector can be written:
(1-20)
Where is the component of the vector along the z axis and is the unit vector of this axis; it’s perpendicular to both and. We have the following relations:
(1-21)
Exercises
16-Three displacements are A = 200 m, due south; B =250 m, due west; C = 150 m, 30.0° east of north. Construct a separate diagram for each of the following possible ways of adding these vectors: R1 = A + B + C; R2 = B + C + A; R3 = C + B + A.
30- Vector A has x and y components of -8.70 cm and 15.0cm, respectively; vector B has x and y components of 13.2 cm and -6.60 cm, respectively. If A - B + 3C = 0, what are the components of C?
43- Given the displacement vectors A = (3ˆi - 4ˆj + 4ˆk) m and B = (2ˆi + 3ˆj -7ˆk) m, find the magnitudes of the vectors (a) C = A + B and (b) D = 2A - B
50-If A =(6.00ˆi - 8.00ˆj ) units, B = (-8.00ˆi + 3.00ˆj ) units, and C = (26.0ˆi +19.0ˆj) units, determine a and b such that a A + b B + C = 0.
**- ; ;
a)Find the magnitude of
b)Determine the sinus of the angle between and
c)Calculate Q=
1