Chapter2.                                            VECTORS

 

2.1       Introduction:  There are some physical quantities, which are completely described by a single number with a unit, for example; mass of a body. On the other hand, there are other quantities that need a direction specified along with the magnitude for a complete description.

The quantities which are described only via magnitudes are called Scalar quantities. For example if we say that we drive 10 miles, we are talking about the total distance traveled. Here we are talking about magnitude only (distance) i.e. a Scalar quantity.

Those quantities which are fully described by magnitude as well as direction are known as Vector quantities. For example, traveling with a velocity of 20 miles per hour due south is not the same as traveling with a velocity of 40 miles per hour due east. These quantities are Vectors and it is important to distinguish them from scalars (physical quantities described by a single number). Examples of vector quantities are forces, velocity or the position of a robot etc.

 

Variables that are vectors will be indicated with a boldface variable, although it is common to see vectors denoted with small arrows above the variable.

 

2.1.1    Unit Vector:    A unit vector is a vector that has a magnitude of one. A vector representing a unit vector is also boldface, and it will have a carat (^) above it to indicate the unit nature of the variable. The unit vector x, when written with a carat, is read as "x-hat" as the carat looks like a hat on the variable.

2.1.2    Zero Vector:   The zero vector, or null vector, is a vector with a magnitude of zero. It is written as 0.

2.1.3    Parallel Vectors: Vectors are parallel if they have the same direction. Both components of one vector must be in the same ratio to the corresponding components of the parallel vector.

 

2.1.4        Collinear Vectors:  Two vectors that are parallel to each other are called ``collinear'', as they can be drawn onto the same line.

 

2.1.5    Anti-parallel vectors:   When two vectors V₁ and V₂  are in opposite directions, their magnitudes being same or not, we say that they are anti-parallel. If they have the same magnitude, the relation between the two vectors is

 

 V₁ = −V₂ or

 V₂ = −V₁

           

2.2:      Magnitude of a Vector: The magnitude of a vector quantity is usually represented by the same letter used for the vector, but in italic with no arrow on top. An alternative notation is the vector symbol with vertical bars on both sides. The two notations are shown below:

 

                                    (Magnitude of V) = V = |V|

 

By definition the magnitude of a vector quantity is a scalar quantity (a number) and is always positive.

If                          r = (x, y, z)

 

represents the vector displacement of point R from the origin, what is the distance between these two points? In other words, what is the length, or magnitude,

 

r = |r|,

 

of vector r. It follows from a 3-dimensional generalization of Pythagoras' theorem that

 

                                                r = (x²+y²+z²)^1/2

Note that if

r = r₁ + r₂

then ,

|r|  ≤ | r₁|+ |r₂|

 

In other words, the magnitudes of vectors cannot, be added algebraically. The only exception to this rule represented by the equality sign in the above expression) occurs when all the vectors in question point in the same direction.

2.3:      Direction of Vectors:   The direction of a vector is often expressed as an angle of rotation of the vector about its "tail" from east, west, north, or south. For example, a vector can be said to have a direction of 30 degrees North of West that means, a vector pointing west has been rotated through 30 degrees towards the north direction, of 65 degrees East of South which means a vector pointing South has been rotated 65 degrees towards the east direction.

The direction of a vector is often expressed as an counterclockwise angle of rotation of the vector about its "tail" from due East. Using this convention, a vector with a direction of 30 degrees is a vector which has been rotated 30 degrees in a counterclockwise direction relative to due east. A vector with a direction of 160 degrees is a vector which has been rotated 160 degrees in a counterclockwise direction relative to due east. A vector with a direction of 270 degrees is a vector which has been rotated 270 degrees in a counterclockwise direction relative to due east. This is one of the most common conventions for the direction of a vector and will be utilized throughout this unit.

                                                         N

 

 

 

 

 

 

                                                                        30^o

                 W                                                                           E

 

 

 

 

 

 

 

                                                        S

 

 

 

30^o counter-clockwise rotations from east.

 

 

 

 

 

 

 

 240^o

 

 

 

 

 

 

 

 

 

 

 

 

 

240^o counter-clockwise rotations from east

 

           

 

2.4.      Components of Vectors:   Vectors are generally oriented on a two dimensional coordinate system. The Cartesian plane has a horizontal axis, labeled X and a vertical axis labeled Y. Vectors in multiple-dimension coordinate systems can be split up into their component vectors. In the two-dimensional case, this results in a X-component and a Y-component.

 

 

 

 

 

 

                                                   F

                                    F_x

 

                                                θ

                                               

                                                            F_y

 

 

 

The figure shows an example of a Force vector (F) broken into its components (F_x & F_y). When  we break a vector into its components, the vector is a sum of the components:

 

F = F_x + F_y

 

To determine the magnitude of the components, we use the rules of triangles

Taking the angle between X-axis and the vector and considering the right triangle including the angle θ, we see that F_x is the adjacent side, F_y is the opposite side, and F is the hypotenuse.

So, from the rules for right triangles, we know that:

 

F_x / F = cosθ

and

F_y / F = sinθ

 

That gives,

 

F_x = F cosθ

and

 F_y = F sinθ

 

 

2.5.      Addition of Vectors:   Two vectors can be added together to determine the result (or resultant).

Let us suppose that an element undergoes a displacement A, followed by a second displacement B. The final result is the same as if the particle had started at the same initial point and undergone a single displacement C.

 In vector addition tail of the second vector is at the head, or tip, of the first vector as shown in the figure                                       

 B

 

 

                         A

 

                                                 C

 

 

 

 

 

We call displacement C the vector sum, or resultant, of displacements A and B.  The relation is represented as;

C = A + B

 

 

 

 

 

 

 

 

 

If we reverse the order of displacements  A and B, with B first and A second, the result remains the same.

 

 

 

 

 

                                     C                   A

 

 

 

                                   B

Thus

 

C = B + A

And

 A + B = B + A.

 

This shows that the, vector addition obeys the commutative law.

 

 

 

 

When vectors A and B are both drawn with their tails at the same point, vector C is the diagonal of a parallelogram constructed with A and B as two adjacent sides.

 

 

 

 

 

The Pythagorean Theorem

 

The Pythagorean Theorem is used for determining the result of addition two vectors which make a right angle to each other. The method is not applicable for adding more than two vectors or for adding vectors which are not at 90-degrees to each other. The Pythagorean Theorem is a mathematical equation which

relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                                                                  c

 

                                        b

 

 

 

 

                                                            a

 

 

 

                                                c² = a² + b²

 

 

2.6.      Subtraction of Vectors:  Subtracting a vector is the same as adding its negative. The difference of the vectors a and b is the sum of a and –b

 

a – b = a + (–b)

 

For example; consider the two vectors V₁ and V₂ shown below:

 

 

                                                 V₁

 

                                                          V₂

 

To subtract,

 V₁ − V₂

 

 

 first pint out the direction of V₂ and since addition is commutative, place the tail of V₁ at the tip of V₂ and perform the addition as show in the figure below:

 

 

 

 

 

 

 

 

 

 

 

       V₁                         V₁ − V₂

 

 

 

 

                  

       -V₂

 

2.7              Multiplication of Vectors

 

2.7.1    Multiplication by Scalar:        Let a be a vector quantity and ά be a scalar quantity. Then,

 

                                                                        b = ά a

is a vector quantity having direction parallel to a and its magnitude is given by;

 

                                                            |b| = ά |a|

A unit vector n parallel to a is given by;

                                                            n =  a

                                                                   |a|

 

 

 

2.7.1.1  Product of vector and scalar in Cartesian components:

Let ,

            a = a_xi + a_yj + a_zk

similarly,

            b = b_xi + b_yj + b_zk      

and we have,

                        b = ά a

therefore,

                        b_xi + b_yj + b_zk = ά (a_xi + a_yj + a_zk) = ά a_xi + ά a_yj + ά a_zk

 

                       

hence,

                                    b_x = ά a_x ;   b_y = ά a_y ;   b_z = ά a_z

 

 

 

2.7.2    Vector products: Two other powerful methods that provides easy working with vectors are the dot (or inner) product and the cross product. The dot product produces a scalar and the cross product operates only on 3−dimensional vectors and produces another vector.

 

 

 

 

2.7.2.1 Dot Products: The scalar product of two vectors A and B is denoted by

 

A · B.

 

The scalar product is also called the dot product.

To define the scalar product A · B of two vectors A and B, draw the two vectors with their tails at the same point as shown in the figure below.

 

The angle between their directions is φ as shown; the angle φ always lies between 0° and 180°. (As usual, we use Greek letters for the angles.) The figure below shows the projection of vector B onto the direction of A;

 

 

                          B    

                              φ

 

                                     A

 

This projection is the component of B parallel to A and is equal to B cos φ.

We define A · B to be the magnitude of A multiplied by the component of B parallel to A

 

A · B = AB cos φ = |A||B|cos φ    

 

where φ ranges from 0° and 180°.

 

Alternatively, we can define A · B to be the magnitude of B multiplied by the component of A parallel to B,

 

 

 

                  B

 

      Acos φ      

                   φ

 

                         A

 

 

Hence

 A · B = B(A cos φ) = AB cos φ,

 

 

 

 

 

which is the same as the previous equation above.

 

When φ is between 0° and 90°, the scalar product is positive.

When φ is between 90° and 180°, it is negative.

When φ = 90°, A · B = 0. The scalar product of two perpendicular vectors is always zero.

 

For any two vectors A and B,

 AB cos φ = BA cos φ.

This means that

A · B = B · A.

 

The scalar product follows the commutative law of multiplication; the order of the two vectors does not matter.

The scalar product can also be expressed in terms of components as follows:

 

A · B = A_xB_x + A_yB_y + A_zB_z   

 

The dot product is distributive:
                                              a.(b + c) = a.b + a.c
and commutative:
                                                      a.b = b.a

The angles between i, j, and k vectors is π/2 radians (90 degrees) and cos π /2 = 0, we can derive a handy alternative definition: Let,
                                                             u = ai + bj + ck
                                                             v = xi + yj + zk
then,
                                              u.v = (ai + bj + ck).( xi + yj + zk)
                      =>u.v = (ai + bj + ck). xi + (ai + bj + ck).yj + (ai + bj + ck).zk

The angle between any nonzero vector and iteself is 0, and cos 0 = 1, so i.i = 1 etc., Hence,

                                                             u.v = a x + b y + c z

This means that for any vector, a,
                                                                     a. a = a²

 

2.7.2.2 Angle between two vectors:  Let the coordinates of any two nonzero vectors u and v . The angle q between them:
 

u = ai + bj + ck
v = xi + yj + zk
u.v = u v cos q
u.v = a x + b y + c z

therefore,
                                                u v cos q = a x + b y + c z

                                               q = cos^-1 o (a x + b y + c z) / ( u v ) p

 

 

2.7.2.3 Cross Product:             The vector product of two vectors A and B, also called the cross product, is denoted by A X B.

The symbol used to represent cross product is a cross (×). Since this product has both,magnitude and direction, it is also known as the vector product.

A × B = AB sin θ ˆn

The vector ˆn ("n hat") is a unit vector perpendicular to the plane formed by the two vectors. The direction of ˆn is determined by the right hand rule.

The cross product is distributive i.e.

 A × (B + C) = (A × B) + (A × C)

but not commutative i.e.

A × B = −B × A

On reversing the order of cross multiplication the direction of the product also get reversed

If we know the components of A and B, we can calculate the components of the vector product, using a procedure similar to that for the scalar product.

 

The components of

C = A X B

 

 

are given by

C_x = A_x + B_x,    

C_y = A_y + B_y,    

C_z = A_z + B_z     

 

The cross product of any vector with itself is zero .i.e.

A × A = 0

Applying this corollary to the unit vectors means that the cross product of any unit vector with itself is zero.

î × î = ĵ × ĵ = ˆk × ˆk = (1)(1)(sin 0°) = 0

It should be noted that the cross product of any unit vector with other will give a magnitude of one, because

sine90° = 1

The right hand rule for cross multiplication relates the direction of the two vectors with the direction of their product. Since cross multiplication is not commutative.

1. Hold your right hand flat with your thumb perpendicular to your fingers. Don’t bend your thumb
2. Point your fingers towards the direction of the first vector.
3. Direct your palm so that when you fold your fingers they point in the direction of the second vector.
4. Your thumb is pointing in the direction of the cross product.