Vectors Parallelogram Law,Triangle Law and Applications

If two vectors are having equal magnitude and certain angle between them , we can find the resultant of the two vectors using the parallelogram law as shown. Using the same concept it can be proved that, if the resultant of two equal and vectors is equal to any one of the vectors then the angle between them is 120°.

Vectors subtraction is similar to that of the vector addition the only differences will be getting an extra negative sign. We can solve all the problems of vectors subtraction using the same concepts of vector addition. All rules like parallelogram law and triangular law can be applied to this concept by taking care of proper signs.

Problem

Let us consider a circular disc of radius R and it is having a translator motion.Find the magnitude and displacement of completion of the have the revolution. Also find the angle made by the resultant.

Lamis Theorem:

This theorem helps in solving some problems vectors. The concept of the theorem and application is solved as explained below. As per this concept, if the vectors are acting and the is in equilibrium, the ratio of the vectors and its opposite sin angle is always constant.

The problem is solved here actually can be solved even with the concept of resolution vectors.

Application of triangular law:

Determination of an external force applied horizontally on the Bob so that the pendulum gets a vertical displacement

Let us consider spherical body having mass m attached to a string of length l two a rigid support as shown. Letters apply a horizontal for F on the spherical body so that it displaces an angle with the vertical. At the instant the spherical body is in a equilibrium position just because there are three forces acting on it as shown. 

By resolving the tension into components we can write a equilibrium equations as shown. Here we can calculate force as well as the tension in terms of the weight of the body.

Problem:

Let us  consider two vectors  x and square root of 2 are having some angle between them. Let the resultant of the two vectors is  square root of six and is perpendicular to the vector X . Then find the value of the x and also find the angle between the two vectors ?

In solving this problem we need to take parallelogram law vectors into consideration. It is considered the two vectors is the two sides of the parallelogram. As the resultant is perpendicular to the X the angle between the two vectors shall be more than 90°. The solution is as presented below.

Let us  considers another problem. Let a person is walking 10 m east, then 10 m north and then 10 root 2 m Southwest. Find his final displacement.

In solving this problem we have to treat each of the position as vector. I J and K are the unit vectors along the x-axis y-axis  and z-axis . They choose the direction of the vector. Each vector is represented and the third vector resolved into components as shown. The resultant of all the vectors is simply the vector sum of all of them.

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Vectors and their usage in Physics

Vectors and their usage in Physics

Physics is always understood in terms of physical quantities.To understand a physical quantities completely in a particular situation, it is enough for a physical quantity to have only magnitude. This kind of physical quantities are called scalars. Maas and length are some of the simple physical quantities who can be treated like scalars.

But some physical quantities demands both magnitude and direction to understand properly. For example if somebody asks you how I have to come to your home?, you cannot say just drop down at the bus terminal and walk 1 km ,you’ll reach my home. You have to say in detail in which direction the person has to travel after getting down at the bus terminal. Thus the situation demands not only magnitude but also the direction in which he has to travel . This kind of physical quantities which need magnitude as well as direction to understand them clearly are being called as vectors in physics.

As a whole we can divide all physical quantities into two categories as scalars and vectors. Vector not only need to have magnitude and direction but has also satisfy certain rules of vector algebra.We are going to have a detailed look regarding the rules of the vectors in the coming post. So when you specify a physical quantity as a vector ,you  have to specify not only its magnitude but its direction .

Having a magnitude and direction alone cannot qualify physical quantity to be treated as a vector. For example time is a physical quantity who has both magnitude as well as a direction in such a way that it always goes in a forward direction. But this doesn't satisfy laws of vectors and hence it cannot be treated like a vector. Electric current and pressure are some of this kind of physical quantities who have both magnitude and direction but still has to be treated like scalars.

The vectors are generally represented in terms of straight line  having a arrow head to it. The arrow head gives the direction of the vector. There are different types of vectors.

The two vectors who are parallel to each other are being treated as parallel vectors are like vectors.

The two vectors who are antiparallel to each other are called unlike vectors or anti parallel vectors.

The two vectors who has same magnitude and direction are treated as equal vectors.

The negative vector of a vector is a vector who has the same magnitude but the opposite direction of the original vector.

A unit vector is a vector who has only a magnitude of one unit but the direction of the original vector.

Resolution of the vectors:

A vector can be only in one direction or in between the two axis. In this case the vector can be resolved into components to identify the part of the vector along x-axis as well as the y-axis. This kind of dividing the vector into parts is called as the resolution of a vector. But when you add the two components of the vector you’re supposed to get back your wards will vector. Then only we can say we have resolved the vector properly. The components of the vectors are also vectors having a specific direction.The components cannot have a direction that is different from the original vector.

 Laws of vectors:

Being a vector, a physical quantity shall satisfy certain laws. The following are the some of the vector laws of addition.

1. Vector addition obeys committee to law.
2. Vector addition obeys associate to law.
3. Vector addition also obeys distributive law.
4. Vectors also satisfy triangle law.

Definition of triangle law:

If three vectors are acting on a point and the point is in equilibrium ,then the three vectors can be represented as three sides of a triangle taken in an order.

Parallelogram law:

if two vectors are represented as two adjacent sides of a parallelogram, then the resultant of the two vectors is the diagonal of the parallelogram passing to the same point.

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