CHAPTER 3 - VECTORS
In our study of physics, we often need to work with physical quantities that have both numerical and directional properties. As noted in section 2.1, quantities of this nature are vector quantities. This chapter is primarily concerned with vector algebra and with some general properties of vector quantities. We discuss the addition and subtraction of vector quantities, together with some common applications to physical situations.
Vector quantities are used throughout this text, and it is therefore imperative that you master both their graphical and their algebraic properties.
3.1 COORDINATE SYSTEM
Many aspects of physics involve a description of a location in space. The mathematical description of an object’s motion requires a method for describing the object’s position at various times. This description is accomplished with the use of coordinates. There are two kinds of represent a point:
a. Cartesian coordinates
b. Polar coordinates
Cartesian coordinates are also called rectangular coordinates but sometimes it is more convenient to represent a point in a plane by its plane polar coordinates (r, θ)
In this polar coordinate system,
r = is the distance from the origin to the point having Cartesian coordinates
θ = is the angle between a line drawn from the origin to the point and a fixed
This fixed axis is usually the positive x axis, and θ is usually measured counterclockwise from it.
From the picture, we find that sin θ = y/r and that cos θ = x/r. Therefore, we can use these equation:
x = r cos θ
y = r sin θ
3.2. VECTOR AND SCALAR QUANTITIES
In Physics we need to work with physical quantities that have both numerical and directional properties. There are two kinds of quantities :
§ Scalar quantities : is completely specified by a single value with an appropriate unit and has no direction or also can defined as a quantity that has magnitude only.
For example distance, speed,length, mass, time, temperature, energy, volume
§ Vector quantities : is completely specified by a number and appropriate units plus a direction.
For example displacement, velocity, force, momentum, acceleration
Example for the vector quantity is displacement.
This picture is represent a particle that move from some point A to some point B. We represent the displacement by an arrow, where some point A as the intial point anf some poin B as the final point. The direction of arrowhead is the direction of displacement, and the length of the arrow is represent the magnitude of displacement.
If the particle travels along some other path from some point A to some point B, such as the broken line in picture, its displacement is still the arrow drawn from some point A to some point B. Displacement depends only on the initial and final positions, so the displacement vector is independent of the path taken between these two points.
3.3 Some Properties of Vectors
3.3.1 Equality of Two Vectors
Two vectors may be defined to be equal if they have the same magnitude and point in the same direction even though they have different starting points. That is A = B only if A = B and if A and B point in the same direction along parallel lines ( picture 3.1 ). This property allows us to move a vector to a position parallel to itself in a diagram without affecting the vector.
3.3.2 Adding Vectors
The rules for adding vectors are conveniently described by graphical methods. To add vector B to vector A, first draw vector A on graph paper, with is magnitude represented by a convenient length scale, and then draw vector B to the same scale with its tail starting from the tip of A. The resultant vector R = A + B is the vector drawn from the tail of A to the tip of B ( Picture 3.2 ).
A geometric construction used for more than two vectors. The resultant vector is R = A + B + C + D. In other words, R is the vector drawn from the tail of the first vector to the tip of the last vector ( Picture 3.3 ).
Picture 3.3 The case of four vectors
When two vectors are added, the sum is independent of the order of addition. Commutative law of addition : A + B = B + A ( Picture 3.4 )
When three or more vectors are added, their sum is independent of the way in which the individual vectors are group together. Associative law of addition : ( A + B ) + C = A + ( B + C ) ( Picture 3.5 )
3.3.3 Subtracting Vectors
The operation of vectors subtraction can be deem as the usual adding vector by reverse the direction of a vector. A – B = A + ( - B )
3.3.4 Multiplying a Vector by a Scalar
Accord from the title, multiplying vector by a scalar will give result a scalar magnitude.
3.4 Components of a Vector and Unit Vectors
(a) a vector A lying in the xy plane can be represented by its component vectors Ax and Ay.
(b) the y component vector Ay can be moved to the right so that it adds to Ax. The vector sum of the component vectors is A. These three vectors from a right triangle.
From figure 3.8 and the definition of sine and cosine, we see that Ax = A cos θ and Ay = A sin θ
These components form two sides of a right triangle with a hypotenuse of length A. Thus, it follows that the magnitude and direction of A are related to its components.
Note : the signs of the components Ax and Ay depend on the angle.
3.4.1 Unit Vectors
A unit vector is a dimensionless vector having a megnitude of exactly. We shall use the symbols i, j, and k to represent unit vectors pointing in the positive x, y, and z directions, respectively.
The produce of the component Ax and the unit vector i is the vector Axi. Likewise, Ayj is a vector of magnitude |Ay| lying on the y axis which representation of vector Ay. Thus, the unit-vector notation for the vector A is A = Axi + Ayj
r = xi + yj
Now, to add vector B to vector A, where vector B has components Bx and By. All we do is add the x and y components seperately. The resultant vector R = A + B is therefore
R = ( Axi + Ayj ) + ( Byi + Byj )
R = ( Ax + Bx )i + ( Ay + By )j
Because R = Rxi + Ryj so
Rx = Ax + Bx
Ry = Ay + By
At times, we need to consider situations involving motion in three component directions. The extension of our methods to three-dimensional vectors is straightforward. If A and B both have x, y, and z components, we express them in the form
A = Axi + Ayj + Azk
B = Bxi + Byj + Bzk
The sum of A and B is
R = (Ax + Bx)i + (Ay + By)j + (Az + Bz)k
Unknown : A = (2.0i + 2.0j)m B = (2.0i – 4.0j)m
Question : Find the sum of two vectors A and B lying in the xy plane
Answer : R = A + B = (2.0 + 2.0)i m + (2.0 – 4.0)j m
= (4.0i – 2.0j)m
T H E R E F O R E
Rx = 4.0 m Ry = -2.0 m
= 4.5 m