Showing posts with label vector addition. Show all posts
Showing posts with label vector addition. Show all posts

Parallelogram law and addition of vectors Video Lesson

We are going to deal about parallelogram law of vectors and using the law to find the addition and subtraction of the given vectors. Vector is a physical quantity that has both magnitude and direction. When we add scalars, we only need to worry about their magnitude,  But adding vectors is little complicated when compared with scalars. We can add them using a basic graphical method. Here we need to shift the second vector in parallel so that the magnitude and direction remains same. Then the tail of the first vector has to be joined with head of second vector to get the resultant of the two vectors. This is little graphical and performing this method is lengthy process when multiple vectors are involved.

The alternate method to add the two vectors is algebric method using parallelogram law of vectors. According to the law, if two vectors are represented as two adjusent sides of a parallelogram starting from the same point,  then the resultant of the two vectors is the diagonal of the parallelogram and its direction also can be found as shown in the video below.


Resultant of two vectors

We can apply the above mentioned law for different cases. What will be the resultant of the two vectors depends on the magnitude of the two vectors, the angle between them. Here in the below video, we are solving different basic possible cases and we have found the resultant of the given two vectors using the parallelogram law of the vectors.


Addition of the two vectors

We also would like to consider the addition of the two vectors using algebric method. Here we represent the vectors with components along the X,Y and The Z axis. When we are adding the vectors, we add the respective components and the find the resultant vector and its direction as shown in the video below.


Relative Velocity

Relative velocity is different from resultant velocity. Relative velocity is the comparative velocity of one body with respect to the other.  A body will have relative velocity only when it has effective displacement when compared with the other body. In the following video, it is explained the way of measuring the relative velocity in brief.

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Vector resolution and Laws of Vectors Video lesson

We are here discussing regarding the introduction of vectors, resolution of vectors and laws of vectors. Vector is a physical quantity that has magnitude, direction and satisfy the laws of vector algebra. Some physical quantities in physics need both magnitude and direction to explain them completely and that kind of vectors are treated as vectors. Simple examples of vectors are displacement,velocity,momentum,force and torque.

Vectors are graphically represented as an arrows and the head of the arrow is treated as its direction. The size of the arrow is directly proportional to the magnitude of the vector. The direction of the vector is identified with its unit vector and it has only direction of the vector and the magnitude of the unit vector is one unit only. Thus any vector can be represented as the product of magnitude of the vector and its unit vector.

The direction of the vector are represented as unit vectors i,j and k along X,Y and Z axis. All of they are having a magnitude of one unit. If two vectors are having same magnitude and same direction then the two vectors are called unit vectors. If their magnitude is same but the direction is opposite, then one vector is called negative vector of the other vector and vice versa. Here is a video lesson on the basics of the vectors.


Resolution of the vector

A vector could be only along one direction and then identifying the direction is easy like it is along X or Y axis. If any vector is in a plane making some angle with any of the axis, then we can not say that it is either along the X or Y axis. To know how much part of the given vector is along the given axis, we can resolve the vector into components. A component is a part of a given vector along a specified direction and we use trigonometry  to resolve the vector into components. If we add the two vectors, we will get back the original vector without any loss. It is explained in the video lesson as shown in the diagram below.

Laws of vectors

Vectors addition and subtraction can be done either graphically or algebraically. For graphical addition or subtraction, we are going to shift the given and required vector parallel. As shifting the vector in parallel  won't change either its magnitude or direction, the vector remains same. Any way this can be done with much ease using mathematical tools. Vector addition satisfies commutative law, and distributive law. But vector subtraction does not satisfy the commutative law. It is explained as shown in the video lesson below.

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Vectors Problems and Solutions Four

In continuation to the problems with solutions in a topic called vectors, we are going to solve some more problems based on vector addition, parallelogram law, components of the vectors and Lami''s theorem. Lami' theorem is used to find comparison of three vectors and can be tried like a alternate option for triangle law of vectors. 

Any vector can be resolved into components along X,Y and Z axis. To identify them with the particular direction, we can multiply the components with unit vectors along the respective axes like i,j and k. Adding all the components back in vector format gives back the original quantity and hence sanctity of the physical quantity is not disturbed. 

It is given in the problem that a man of mass 80 kilogram is supported by two cables as shown and we need to know the ratio of the tensions in that cables.The problem is as shown in the diagram below.   


Solution

The angles at the two corners of the triangle is given to us as 60 and 30 degree. As we know that the total angle in a triangle is 180 degree, the angle at the third corner is 90 degree. 


According to Lami's theorem, the ratio of the force and its SIN component is constant. By applying that rule, we solve the problem as shown in the diagram below.


Problem

This problem is also a similar one to the earlier. It is given in the problem that two pegs are separated by 13 cm. A body of weight W is suspended using a thread of 17 cm. We need to know the tensions in the strings separated. The problem is as shown in the diagram below.


Solution

It is given in the problem that the total length of the string 17 cm is divided into two parts of 5 and 12 centimeter.The sides 5,12 and 13 are the combination of a right angled triangle. So we can find the angle at the remaining two corners as shown in the diagram below.

Once we know the angles, we can apply Lami's theorem and solve the problem as shown.


Problem

According to the given problem the vector sum of the given to vectors is the third vector and magnitude of the sum of the squares of the same two vectors is the square of the magnitude of the same third  vector and we need to find the angle between the two vectors. The problem is as shown in the diagram below.


Solution

We can write the resultant of  the two vectors using the parallelogram law of the vectors. By further substituting the condition given in the problem, we can solve the problem as shown below.


Problem

It is given in the problem that the sum of two unit vectors is also a unit vector in terms of magnitude. We need to find the difference of the two unit vectors and the problem is as shown in the diagram below.


Solution

 Using the concept the parallelogram law of vectors, we can find the angle between them as shown below. We know that the vectors are unit in magnitude and we can apply this condition.


Problem

It is given that l,m and n are the directional cosines of a vector and we need to find the relation among them. The problem is as shown below.


Solution

We can define direction cosine as the ratio of the component of the vector and the magnitude of the vector. The same can be defined with different components of the vector. Their squares and sum of them as shown below.


 Problem

 A metal sphere is hung by a string fixed to the wall as shown in the diagram. It is pushed away from the wall and different forces acting on it are the weight, tension in the string and the applied force. We need to check which of the given relation is wrong.

Solution

 At the instantanious position,  the sphere is in the equallibriem. Thus the three forces acting on it can be expressed as the three sides of the triangle and their vector sum is zero. We can solve the problem as shown below.

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Vectors Problems and Solutions Three

We are solving series of problems on relative velocity and its applications. Vector is a physical quantity that has both magnitude,direction and satisfy the laws of vector algebra. Relative velocity is the comparative velocity of one body with the other. We do find relative velocity of one body with respect to other. 

It is given in the problem that relative velocity of him is found to be double to his original velocity. We shall know at what angle he has to hold his umbrella with the vertical so that he is protected from the rain. The problem is as shown in the diagram below.


Solution

We know that the relative velocity of the man with respect to the ground is the vector sum of two vectors. One is the velocity of the man with respect to the ground and the other is the velocity of the rain with respect to the ground. We know that the relative velocity makes some angle with  velocity of the rain with respect to the ground and that is with the vertical. By drawing the diagram, we can solve the problem as shown in the diagram below.


Problem

It is given  in the problem that the water in a river is having a constant velocity and the boat in the river also moves with a constant velocity. The boat starts one point goes downstream and come back to the same point against the water. We need to know the total time of the journey. The problem is as shown in the diagram below.


Solution

When the boat is going along the stream of the river, the water is supporting him to cross the river. Hence he covers the distance in a less time. We know that the distance is the product of effective speed and time. Thus we can measure the time for downstream as shown in the diagram below.


Similarly when the boat is going upstream, it has to go against the river flow and hence it takes a lot of time to cross in comparison. The total journey time is the sum of these two times. The solution to the problem is as shown in the above diagram.

Problem

Two vectors are given in the problem and we need to know the component of one vector with respect to the other. The problem is as shown in the diagram below.


Solution

Any vector can be resolved into components. We know that the dot product is the product of one vector and the component of the other vector along the direction of the first vector. We can find that component using the definition of the dot product. The solution is as shown in the diagram below. The component of the second vector along the first vector is nothing but its horizontal components.


Problem

The radius vector is given having its components along all three directions. Angular velocity of the particle is also given to us and we need to measure the linear velocity of the particle.


Solution

We know that linear velocity can be expressed as the cross product of the radius vector and the angular velocity. Angular velocity is defined as rate of change of angular displacement. Cross product is the product of two vectors where the resultant is also a vector. We can find the product using matrix determinant method  as shown below. The resultant is a vector having only Y component.


Problem

In the given problem, adjacent sides of the parallelogram are given to us and using that data, we need to find the area of the parallelogram. The problem is as shown in the diagram below.


Solution

It can be proved that the magnitude of the cross product of two vectors that are represented as the two adjacent sides of the parallelogram  gives the area of that. Taking that concept into consideration, we can  solve the problem as shown in the diagram below.


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Vectors Problems and Solutions Two

We are solving problems basing on the concepts of vectors and its applications.Vector addition is different from scalar addition. Here we need to add taking not only its magnitude but also its direction. Thus we need to follow parallelogramlaw of vectors to find the resultant of the two vectors.

Displacement, velocity are the simple examples of vectors and speed and distance are the simple examples of hte scalars.

In the given problem a car is weighing 100 kg and it is kept on a slope and the angle of the slope is given to us in the problem. We need to know the component of the weight of the car parallel to the slope. The problem is as shown in the diagram below.



Solution

We can resolve the vector into components. Each component is also a vector and it has specific direction. To identify the direction, we can multiply the component of the vector with unit vectors along X and Y direction. When we add both the components, we can get the original vector and we can find the magnitude and direction of the given vector basing on this concept.


Problem

The problem is regarding relative velocity. It is the comparative velocity of one body with respect to the other. Two ships are moving with certain velocities and we need to find the relative velocity of with respect to the other. The problem is as shown in the diagram below.


Solution

If two bodies are moving in the same direction, then the relative velocity is the difference of velocities  of two bodies. We can find the magnitude and direction as shown in the diagram below.


Problem

This problem is about finding average acceleration of the particle. It is first moving eastwards with a certain velocity and then it has changed its path towards north and continued moving with the same speed for 10 second. We need to find the average acceleration of the particle. The problem is as shown in the diagram below.


Solution

Velocity of the particle along east can be expressed in terms of "i" component and velocity along north can be expressed in terms of "j" component of the vector. They are the unit vectors along X and Y axis.

We can find the difference of the two vectors magnitude and by using the  concept that acceleration is the rate of change of the velocity, we can solve the problem as shown below.


Problem

In this problem, a man in a car is moving with a speed known on a raining day. To protect himself from the rain, he has to hold the umbrella at an angle with the vertical. We need to find the velocity of the rain and the problem is as shown in the diagram below.


Solution

We need to know the direction of the relative velocity of the rain with respect to man. It is the vector sum of the velocity of the rain with respect to ground and the velocity of the ground with respect to the man.

We can draw the direction as shown below and hence, we can solve the problem.


Problem

It is also a similar problem. It is raining vertically with a known velocity and we need to know the direction in which a man see that rain when he is moving horizontally with a known speed. the problem is as shown int the diagram below.


Solution

This problem is also solved in the similar way of the previous problem. We can get the angle and hence the relative velocity as shown in the diagram below.


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Vectors Problems and Solutions One

A Physical quantity that has both magnitude and direction and satisfy  the laws of vector algebra is called a vector. Some physical quantities like displacement and velocity need both magnitude and direction to understand them properly and they were treated like vectors. Here we are solving problems based on this vector concepts. The first problem tells that the vector resultant of two vectors is given for us and we need to find the angular separation between the given vectors. The problem is as shown in the diagram below.


Solution

The resultant of two vectors can be found using the parallelogram law of vectors. According to it, if two vectors are represented as the two sides of a parallelogram from one common point, then the resultant of that two vectors is the diagonal of the parallelogram from the same point.

Applying that concept and making further simplification, we can solve the problem as shown below.


Problem

It is given in the problem that the two vectors are having the same magnitude and they are separated by known angle and they are in the same direction. We need to find the resultant of them. The problem is as shown in the diagram below.


Solution

Again we are using the Parallelogram law of the vectors. By applying the trigonometry rules, we can solve the problem as shown in the diagram below.


Problem

The next problem is also the same and we need to find the difference but not the sum in the given problem. It is as shown in the diagram below.


Solution

Even the solution is same. The only difference is,we need to find the difference but not the sum of the two vectors. Thus while we are applying the parallelogram law, we need to  apply the angle as the difference of one eighty degree and the given angular separation of the two vectors. Thus we are able to find the answer as shown in the diagram below.


Problem

It is given in the problem that eleven forces are acting on a point and each force makes an angle 30 degree with its neighbor. We need to find the resultant force acting on the particle. The problem is as shown below.


Solution

Force is a physical quantity that changes are try to change the state of a body. It is a vector that has both the magnitude and direction. In the given problem, out of eleven forces, ten forms five pairs such that each force has another force of equal magnitude but opposite direction. Thus they cancel each other. Hence only one force is remaining and that itself is the resultant. The solution is as shown below.


Problem

It is given in the problem that the sum of the two vectors is 16 newton. If the resultant force is eight newton and it is perpendicular to the smaller force, we need to measure two forces involved here.


Solution

By using the definition of the parallelogram law both in terms of magnitude and direction and by simplifying it further, we can solve the problem as  shown in the diagram below.


 Problem

 It is given in the problem that a weight has been suspended from the mid point of a rope connected between the two points of the same level. The rope lost its horizontalness and we need to know the minimum tension required to make to horizontal. The problem is as shown in the diagram below.

Solution

 The weight appliyed makes the string non horizontal and it generates a tension in it. We can resolve it into components as shown. The sum of vertical components and it has to be equal to the weight. By solving the equation, we can get the answer as infinite.


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