## Saturday, November 5, 2016

### 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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