Basic Concepts Of Vectors

Basic Concepts Of Vectors: Vectors And Equilibrium: Physical quantities that have both numerical and directional properties are called vectors. This chapter is concerned with the vector algebra and its applications in problems of equilibrium of forces and equilibrium of torques.

Basic Concepts Of Vectors

vectors

As we have studied in school physics, there are some physical quantities which require both magnitude and direction for their complete description, such as velocity, acceleration and force. They are called vectors. In books, vectors are usually denoted by bold face characters such as A, d, r and v while in handwriting, we put an arrowhead over the letter e.g. d. if we which to refer only to the magnitude of a vector d we use light face type such as d.

Rectangular Coordinate System

Two reference lines drawn at right angles to each other as shown in Fig. 2.1 (a) are known as coordinate axes and their point of intersection is known as origin. This system of coordinate axes is called Cartesian or rectangular coordinate system.

One of the lines is named as x-axis and the other the y-axis. Usually the x-axis is taken as the horizontal axis, with the positive direction to the right, and the y-axis as the vertical axis with the positive direction upward.

The direction of a vector in a plane is denoted by the angle which the representative line of the vector makes with positive x-axis in the anti-clock wise direction, as shown in Fig 2.1 (b). the point P shown in Fig 2.1 (b) has coordinates (a,b). this notation means that if we start at the origin, we can reach P by moving ‘a’ units along the positive x-axis and then ‘b’ units along the positive y-axis.

positive_x-axis_coordinates_(a,b) — Fig. 2.1 (b)

The direction of a vector in space requires another axis which is at right angle to both x and y axes, as shown in Fig 2.2 (a). the third axis is called z-axis.

The direction of a vector in space is specified by the three angles which the representative line of the vector makes with x, y and z axes respectively as shown in Fig 2.2 (b). the point P of a vector A is thus denoted by three coordinates (a, b, c).

x, y and z axes_respectively — Fig. 2.2 (b)

Addition Of Vectors

Given two vectors A and B as shown in Fig 2.3 (a), their sum is obtained by drawing their representative lines in such a way that tail of vector B coincides with the head of the vector A. now if we join the tail of A to the head of B, as shown in the fig. 2.3 (b), the line joining the tail of A to the head of B will represent the vector sum (A+B) in magnitude and direction. The vector sum is also called resultant and is indicated by R. thus R = A+B this is know as head to tail rule of vector addition. This rule can be extended to find the sum of any number of vectors. Similarly the sum B + A is illustrated by black lines in fig 2.3 (c). the answer is same resultant R as indicated by the red line. Therefore, we can say that

A + B = B + A ………….. (2.1)

So the vector addition is said to be commutative. It means that when vectors are added, the result is the same for any order of addition.

Resultant Vector

The resultant of a number of vectors of the same kind-force vectors for example, is that single vector which would have the same effect as all the original vectors taken together.

Vector Subtraction

The subtraction of a vector is equivalent to the addition of the same vector with its direction reversed. Thus, to subtract vector B from vector A, reverse the direction of B and add it to A, as shown in Fig. 2.3 (d).

A – B = A + (-B) where (-B) is negative vector of B

Basic Concepts Of Vectors: (Vi) Multiplication Of A Vector By A Scalar: The product of a vector A and a number n > 0 is defined to be a new vector nA having the same direction as A but a magnitude n times the magnitude of A as illustrated in Fig. 2.4. if the vector is multiplied by a negative number, then its direction is reversed.

In the event that n represents a scalar quantity, the product nA will correspond to a new physical quantity and the dimensions of the two quantities which were multiplied together. For example, when velocity is multiplied by scalar mass m, the product is a new vector quantity called momentum having the dimensions as those of mass and velocity.

Unit Vector

A unit vector in a given direction is a vector with magnitude one in that direction. It is used to represent the direction of a vector.

A unit vector in the direction of A is written as Â, which we read as A hat, thus

The direction along x, y and z axes are generally represented by unit vectors î, ĵ and k̂ respectively (fig. 2.5 a). The use of unit vectors is not restricted to Cartesian coordinate system only. Unit vectors may be defined for any direction. Two of the more frequently used unit vectors are the vector r̂ which represents the direction of the vector r (fig. 2.5 b) and the vector n̂ which represents the direction of a normal drawn on a specified surface as shown in Fig 2.5 (c)

Null Vector

Null vector is a vector of zero magnitude and arbitrary direction. For example, the sum of a vector and its negative vector is a null vector.

A + (-A) = 0 …………….. (2.2)

Equal Vectors

Two vectors A and B are said to be equal if they have the same magnitude and direction, regardless of the position of their initial points.

This means that parallel vectors of the same magnitude are equal to each other.

Rectangular Components of a Vector

A component of a vector is its effective value in a given direction. A vector may be considered as the resultant of its component vectors along the specified directions. It is usually convenient to resolve a vector into components along mutually perpendicular directions. Such components are called rectangular components.

Let there be a vector A represented by OP making angle θ with the x-axis. Draw projection OM of vector OP on x-axis and projection ON of vector OP on y-axis as shown in Fig.2.6. projection OM being along x-direction is represented by and projection ON = MP along y-direction is represented by . by head and tail rule

vector-rectangular-components — Fig. 2.6

Thus and are the components of vector A. since these are at right angle to each other, hence, they are called rectangular components of A. considering the right angled triangle OMP, the magnitude of or x-component of A is

Determination Of A Vector From Its Rectangular Components

If the rectangular components of a vector, as shown in Fig. 2.6, are given, we can find out the magnitude of the vector by using Pythagorean theorem.

In the right angled ∆ OMP,

Position Vector

The position vector r is a vector that describes the location of a point with respect to the origin. It is represented by a straight line drawn in such a way that its tail coincides with the origin and the head with point P (a,b) as shown in fig. 2.7(a). the projections of position vector r on the x and y axes are the coordinates a and b and they are the rectangular components of the vector r. hence

in there dimensional space, the position vector of a point P (a,b,c) is shown in fig. 2.7 (b) and is represented by

Example 2.1: the positions of two aeroplanes at any instant are represented by two points A (2, 3, 4) and B(5, 6, 7) from an origin O in km as shown in Fig. 2.8.

(i) What are their position vectors?

(ii) Calculate the distance between the two aeroplanes.

Solution: (i) A position vector r is given by

r = a î + b ĵ + 4 k̂

Thus position vector of first aeroplane A is

OA = 2 î + 3 ĵ + 4 k̂

And position vector of the second aeroplane B is

OB = 5 î + 6 ĵ + 7 k̂

By head and tail rule

OA + AB = OB

Therefore, the distance between two aeroplanes is given by

AB = OB – OA = (5 î + 6 ĵ + 7 k̂) – (2 î +3 ĵ + 4 k̂)

= ( 3 î + 3 ĵ + 3 k̂ )

Magnitude of vector AB is the distance between the position of two aeroplanes which is then:

Vector Addition by Rectangular Components

Let A and B be two vectors which are represented by two directed lines OM and ON respectively. The vector B is added to A by the head to tail rule of vector addition (fig 2.9). thus the resultant vector R = A + B is give, in direction and magnitude, by the vector OP.

In the Fig 2.9 , and are the x components of the vectors A, B and R and their magnitudes are given by the lines OQ, MS, and OR respectively, but

Which means that the sum of the magnitudes of x-components of two vectors which are to be added, is equal to the x-component of the resultant. Similarly the sum of the magnitudes of y-components of two vectors is equal to the magnitude of y-component of the resultant, that is

Since and are the rectangular components of the resultant vector R, hence

The magnitude of the resultant vector R is thus given as

And the direction of the resultant vector is determined from

Similarly for any number of coplanar vectors A, B, C…….. we can write

Do you know?

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The vector addition by rectangular components consists of the following steps.

Find x and y components of all given vectors.
Find x-component of the resultant vector by adding the x-component of all the vectors.
Find y-component of the resultant vector by adding the y-components of all the vectors.
Find the magnitude of resultant vector R using

Find the direction of resultant vector R by using

where is the angle, which the resultant vector makes with positive x-axis. The signs of and determine the quadrant in which resultant vector lies. For that purpose proceed as given below.

Irrespective of the sign of , determine the value of from the calculator or by consulting trigonometric tables. Knowing the value of ф, angle θ is determined as follows.

If both are positive, then the resultant lies in the first quadrant and its direction is θ = ф.
If is – ive and is +ive, the resultant lies in the second quadrant and its direction is θ = 180⁰ – ф.
If both are –ive, the resultant lies in the third quadrant and its direction is θ = 180⁰ – ф.
If is positive and is negative, the resultant lies in the fourth quadrant and its direction is θ = 360⁰ – ф.

Example 2.2: two forces of magnitude 10 N and 20 N act on a body in direction making angles 30⁰ and 60⁰ respectively with x-axis. Find the resultant force.

Solution:

Step (i) x-components

The x-component of the first force : cos 30⁰

= 10 N x 0.866 = 8.66 N

The x-components of second force =

= 20 N x 0.866 = 17.32 N

Step (ii)

The magnitude of y component of the resultant force F

Step (iii)

The magnitude of y component of the resultant force F

Step (iv)

The magnitude F of the resultant force F

Step (v)

F the resultant force F makes and angle θ with the x-axis then

Example: 2.3: find the angle between two forces of equal magnitude when the magnitude of their resultant is also equal to the magnitude of either of these forces.

Solution: let θ be the angle between two forces F1 is along x-axis. Then x-component of their resultant will be

And y-component of their resultant is

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Product Of Two Vectors

There are two types of vector multiplications. The product of these two types are known as scalar product of two vector quantities is a scalar quantity, while vector product of two vector quantities is a vector quantity.

Scalar or Dot Product

The scalar product of two vectors A and B is written as A.B and is defined as

A.B = AB cos θ …………………… (2.17)

Where A and B are the magnitudes of vectors A and B and θ is the angle between them.

For physical interpretation of dot product of two vectors A and B, these are first brought to a common origin (fig. 2.10 a).

scalar_or_dot_product_01 — Fig. 2.10 (a)

scalar_or_dot_product_02 — Fig. 2.10 (b)

Then, A.B = (A) (projection of B on A)

A.B = A (magnitude of component of B in the direction of A)

= A (B cos θ) = AB cos θ

Similarly B.A = B (A cos θ) = BA cos θ

We come across this type of product when we consider the work done by a force F whose point of application moves a distance d in a direction making an angle θ with the line of action of F, as shown in Fig. 2.11.

Work done = (effective component of force in the direction of motion) x distance moved

= (F cos θ) d = Fd cos θ

Using vector notation

F.d = Fd cos θ = work done

Characteristics of scalar product

Since A.B = AB cos θ and B.A = BA cos θ, hence, A.B = B.A the order of multiplication is irrelevant. In other words. Scalar product is commutative.
The scalar product of two mutually perpendicular vectors is zero. A.B = AB cos90⁰ = 0

In case of unit vectors î , ĵ and k̂, since they are mutually perpendicular, therefore,

Î . ĵ = ĵ . k̂ = î = 0 …………… (2.18)

The scalar product of two parallel vectors is equal to the product of their magnitudes. Thus for parallel vectors (θ = 0⁰)

A.B = AB cos 0⁰ = AB

In case of unit vectors

Î . Î = ĵ. ĵ = k̂. k̂ = 1 …………….. (2.19)

And for antiparallel vectors (θ = 180⁰)

A.B = AB cos 180⁰ = -AB

The self product of a vector A is equal to square of its magnitude.

Scalar product of two vectors A and B in terms of their rectangular components

Equation 2.17 can be used to find the angle between two vectors: since,

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Example 2.4: A force F = 2 î + 3ĵ units, has its point of application moved from point A (1,3) to the point B (5,7) find the work done.

Work done = F.d = (2î + 3ĵ). (4î + 4ĵ)

= 8 + 12 = 20 units

Example 2.5: find the projection of vector A=2î -8ĵ +k̂ in the direction of the vector B = 3î – 4ĵ – 12k̂.

Solution: if θ is the angle between A and B, then A cosθ is the required projection.

Where B̂ is the unit vector in the direction of B

Vector or cross product: The vector product of two vectors A and B, is a vector which is defined as

A x B = AB sinθ n̂ ……….. (2.22)

Where n̂ is a unit vector perpendicular to the plane containing A and B as shown in fig. 2.12 (a). its direction can be determined by right hand rule. For that purpose place together the tails of vectors A and B to define the plane of vectors A and B. the direction of the product vector is perpendicular to this plane. Rotate the first vector A into B through the smaller of the two possible angles and curl the fingers of the right hand in the direction of rotation, keeping the thumb erect. The direction of the product vector will be along the erect thumb, as shown in the fig 2.12 (b). because of this direction rule, B x A is a vector opposite in sign to A x B. hence,

A x B = -B x A …………….. (2.23)

Characteristics of cross product

Since A x B is not the same as B x A, the cross product is non commutative.
The cross product of two perpendicular vectors has maximum magnitude A x B = AB sin90⁰ n̂ = AB n̂ in case of unit vectors, since they form a right handed system and are mutually perpendicular fig. 2.5 (a)

Î x ĵ = k̂ , ĵ x k̂ = î , k̂xî = ĵ

The cross product of two parallel vectors is null vector, because for such vectors θ = 0⁰ or 180⁰. Hence

A x B = AB sin0⁰n̂ = AB sin 180⁰n̂ = 0

As a consequence A x A = 0

Also î x î = ĵ x ĵ = k x k = 0 …………….. (2.24)

Cross product of two vectors A and B in terms of their rectangular components is:

A x B = (AxÎ + Ayĵ+Azk̂)x(Bxî +By ĵ+Bzk̂)

A x B = (AyBzBy) Î + (AzBx – AxBz) Ĵ+(AxBy-AyBx) k̂………..(2.25)

The result obtained can be expressed for memory in determinant form as below:

A x B =

I j k

Ax Ay Az

Bx By Bz

vector_or_cross_product_B x A — Fig. 2.12 (d)

The magnitude of A x B is equal to the area of the parallelogram formed with A and B as two adjacent sides (Fig. 2.12 d).

Examples of vector product

When a force F is applied on a rigid body at a point whose position vector is r from any point of the axis about which the body rotates, then the turning effect of the force, called the torque

τ = r x F

The force on a particle of charge q and velocity v in a magnetic field of strength B is given by vector product.

F = q (v x B)