Scalar Product and Vector Product of Vectors

Vector multiplication has two possibilities. If two vectors are multiplied and the resultant is a scalar,then that kind of a product is called scalar product. While you’re measuring  the scalar product we have to multiply one vector  with the component of the other vector that is acting along the direction of the first vector.

If if two vectors are perpendicular to each other then their  scalar product is null.Basing on this concept we can solve some problems and identify the unknown value when it is given as a condition the problem that the two vectors are perpendicular to each other.

Work done is a simple example of a scalar product which is a product of both force and displacement. It is a scalar because the product of them is not going to have direction but only a magnitude.

Vector product:

If two vectors are multiplied and the resultant is a vector then that kind of a product is called as cross product. The output of this product is not only going to have magnitude and direction and is also going to satisfy the laws.To find a direction of this vector are we can use cork Screw rule or right hand thumb rule.

As per the corkscrew rule, if the head of the screwy is rotating from one vector to other, their cross product  vector is going to move along the direction of the tip of the screw  who is the perpendicular plane of these two vectors.

We can find the value of the cross product using the mathematical Matrix Method has shown.

We can find the area of parallelogram and area of the triangle using the cross product as shown.

Moment of the force can be expressed as a cross product of force and the perpendicular distance. We can explain in detail that how it is going to be a cross product of two vectors are shown below.

Related Posts

Units and measurement
Writing dimensional formula
Errors and approximations

Vectors and their usage in Physics

Parallelogram law,Triangle law and applications

Relative Velocity and Motion of a boat across a river

Relative Velocity and Motion of a Boat across a River

Relative velocity is the comparative velocity of one body with respect to other body. One body will have a relative velocity with respect to other one only when there is a relative motion between them. If two bodies are having the motion in the same direction relative velocity of one with respect to other is the difference between them.It is simply because one bodies having more velocity when compared with the other body and how much more can be obtained only by subtracting that from the other value.

If two bodies are travelling in the opposite direction we can incorporate the same concept but being the bodies are in the opposite direction the resultant will become automatically the some of the two vectors as shown.

If two vectors are having an angle between them to find the relative velocity between them we shall incorporate a third body in between them. Generally the third body is the ground.

Expressing velocity of the any body with respect to ground is more appropriate than expressing the velocity of the ground with respect to other body.It is simply because it is not the ground that is moving but the body moving on the ground.

Problem and Solution

Basing on a relative motion of two bodies. Let us consider two bodies each having the same velocity 10 m/s,one moving along the east other moving along the north from the same point. Find the relative velocity of one body with respect to other body?

While we are solving the problem we need to take a reference and the ground into consideration as reference.After identifying the answer,we can identify the direction of the vector.

Problem

The person is walking in the rain feel that the velocity of the rain is as twice as his velocity. It which angle you should hold his umbrella with vertical if he’s moving in a forward direction and training is happening in a vertical downward direction therefore he cannot be drenched in the rain?

Solution

This problem can also be solved basing on the concept of relative velocity as shown below. Whenever requirement of the third body is there we always get the ground into consideration because it is always there. We prefer to say the velocity of the body with respect to ground than in the  reverse way because bodies move on the ground.

Motion of a boat in a river

There are four different possible pace of about moving across the river.

Case one

When a boat is crossing the river along the direction of the river:

In this case the motion of the boat is bit easy because it is supported by the stream of water therefore the boat takes less time to cross the river.

Case two

Let us consider a case that the boat is moving against the steam of the river. In this case as boat has to overcome the opposition of the river it takes more time.

Case three

Let us consider a case boat has to cross the river in such a way that it has to reach the exact opposite position.In this case we are actually not going along the river but where crossing the river.

If you travel straight to the opposite point as the river pushes you are not going to reach the exact opposite point. That’s why we shall drive river boat with an angle  with the vertical.

In the following derivation we have discussed the that with what angle he shall drive therefore he will be reaching the exact opposite point.In this case you are going to reach the exact opposite point means the path is the shortest but it is going to take the longest time to cross the river.

Case four

In this case boat will go straight to the opposite point and being the river is going to push it it is not going to reach the exact opposite point but some other point in the bank of the river. In this case the party is going to be longest but the time is going to be shortest.

So here we have two choices.When you want to cross the river and the shortest path we have two choose case three where as when you want to cross the river with the shortest time we had to choose case four.

Case five

Suppose you are crossing the river in such a way that you are making an angle  with the vertical, but it is not sufficient to reach the exact opposite position.

In this case we have resolve the component of velocity of the boat along a horizontal and vertical parts. The vertical part will help you to identify the time taken to cross the river whereas the horizontal component of velocity has to be subtracted from the velocity of the river value while calculating the drift of the water. Drift is simply a particular value of the displacement because of which the boat has missed the exact opposite position. The equation for this drift is as shown.

Related Posts

Units and measurement
Writing dimensional formula
Errors and approximations

Vectors and their usage in Physics

Parallelogram law,Triangle law and applications

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.

 Related Posts

Units and measurement
Writing dimensional formula
Errors and approximations

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.

Related Posts

Units and measurement
Writing dimensional formula
Errors and approximations