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

Dot Product and Cross Product of Vectors Video Lesson

Dot product and cross product of two vectors is a way of multiplication of the vectors.Vector is a physical quantity that has both magnitude and direction and satisfies the vector algebra. Some physical quantities demands both magnitude and direction with out which we cannot explain the physical quantity. Vector has to be added to the other vectors  but we can not add one vector with the other vector. Vector subtraction also follows the same rules of vector addition with the concept of negative vector. 

Vector quantity can be multiplied with a scalar also and the resultant is a vector with the same magnitude of the given vector. A vector can be multiplied with a vector and the resultant can be a scalar or vector. If two vectors are multiplied and the product is a scalar and that kind of multiplication is called scalar product or dot product of two vectors. To get the dot product of two three dimensional vectors, we shall multiply the corresponding components and add all of them.  A video lesson is given below to explain the dot product as shown in the diagram below.


Work done as dot product

We know  that work is said to be done when applied force is able to produce some displacement. When we apply some force at some angle on a body  on a horizontal surface. The total force is not acting along the horizontal surface and only a component of force is acting along the horizontal. To know that value, we need to resolve the force vector into components. The component of the force acting along the direction of the displacement is producing the displacement and the other component has no impact on the displacement. Thus we need to consider only a component of force and that shall be multiplied with the displacement to get the work done. It is explained as the dot product of force and displacement as shown in the video lesson below.


Cross Product of two vectors

If we multiply one vector with another vector and the resultant of the product is a vector then the kind of vector product is called cross product of the given vectors. The resultant will have not only magnitude and also has the direction. To find the cross product of two vectors of two vectors, we need to multiply the cross product of unit vectors and get the resultant of two unit vectors also a unit vector but having the different direction. To get the direction of cross product of the two vectors we need to use right hand thumb rule or cork screw rule as explained in the video lesson below.


Torque as the cross product of two vectors

Torque means the turning effect and it is a physical quantity useful in understanding the rotational motion of a body. We shall apply a force away from axis of rotation. We know that the rotating particle is at a distance from the axis of rotation. Applied force and the perpendicular distance from the axis of rotation both are vectors. The resultant of two vectors product is also a vector and we can find the torque as shown in the video lesson below.


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Rotational Dynamics Problems with Solutions Two

We are solving series of problems based on the concept rotational dynamics. Moment of inertia is a physical quantity that measures how difficult it is to rotate a body about a given axis. This is similar to mass of translatory motion. Moment of inertia can be defined as the summation of product of mass of each particle with  the square of the distance from the axis of rotation. It depends on the mass of the body and axis of rotation of the body. To over come this moment of inertia, we need to apply torque which means turning effect. It is cross product of distance of the body from axis of rotation and the force applied.

Problem

A mass 1.9 kilogram is suspended from a string of length half meter and it is at rest. Another body of mass 100 gram is moving horizontally strikes this mass and sticks to it. If the combined mass is just able to complete the circle, we need to find the initial velocity of the body. Problem is as shown in the diagram below.


Solution

The kind of collision after which both the bodies moves together is called inelastic collision and here both the bodies moves with common velocity. For this combination to move and complete vertical circular motion, it shall have some minimum velocity at the bottom and taking that into consideration, we can solve the problem as shown in the diagram below. We have applied law of conservation of linear momentum to the given data as shown.


Problem

Three point sized bodies each of same mass are fixed at the three corners of of triangle. We need to find the moment of inertia of the system about an axis passing through the center of frame and perpendicular to the frame. Problem is as shown in the diagram below.


Solution

Each particle is at a distance from the axis of rotation and it can be found using basic geometric principles. All of them are identical with axis and hence moment of inertia of the system is the sum of each of them. Problem is solved as shown in the diagram below.


Problem

Radius of gyration of a body is 18 centimeter when it is rotating about about an axis passing the center of mass of the body. If radius of gyration of the same body is 30 cm about a parallel axis to the first axis, then we need to find the perpendicular distance between two parallel axes.


Solution

Moment of inertia of a body depends on the axis of rotation and it changes with that. To find the moment of inertia of one axis that is parallel to another axis, we need to use parallel axes theorem. According to that theorem we can write statement as discussed in the chapter and further problem can be solved as shown in the diagram below.


Problem

Three identical rings each of mass m  and radius r are placed in the same plane such that each one ring is in touch with the other two. We need to find the moment of inertia of  the system about an axis passing through center of any one ring and the axis is perpendicular to the plane. Problem is as shown in the diagram below.


Solution

We know the moment of inertia of a ring about an axis passing through the center and perpendicular to the plane as a standard result. Other ring axis is a parallel axis to the first axis and their distance of separation is 2r. We can apply parallel axes theorem and find the moment of inertia of other two rings and the moment of inertia of the system is sum of all of them. Solution is as shown in the diagram below. 


Problem

The handle of a door is at a distance of 40 centimeter from the axis of rotation. If a force of 5 newton is applied on the handle at a known angle, we need to measure the torque generated and the problem is as shown in the diagram below.


Solution

We know that the torque is turning effect and it is the cross product of force applied to the distance of the particle from the axis of rotation. By applying that concept, we can solve the problem as shown in the diagram 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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