- #1
hellraiser
What is the difference between the component and resolved parts of a vector? To me both seem the same. Can anyone please explain me with a simple example. Thanx.
hellraiser said:I was doing a problem that seemed to make me believe there was a difference. The definition they have given is:
Given a diagonal we can draw infinite number of parallelograms. Each pair of sides will give a pair of components.
That is true. You can resolve a vector into components in infinite number of ways.
If we are given a vector and we find component of vector in given directions such that they are equivalent to the given vector, this is resoultion of vectors.
Here directions are given and hence there is only one way. But if we are the ones who will be selecting the directions, then again there wil be infinite resolutions.
The problem I was doing was
Two vectors P and Q have resultant R. The resolved part of R in direction of P is Q. If A be the angle between the vectors prove that
sin (A/2)=sqrt(P/2Q)
I got the answer. But I still don't understand why I took the resolved part of R in direction of P to be R cos(x) .. x is the angle which R makes with P.
That is true. You can resolve a vector into components in infinite number of wayshellraiser said:I was doing a problem that seemed to make me believe there was a difference. The definition they have given is:
Given a diagonal we can draw infinite number of parallelograms. Each pair of sides will give a pair of components. P.
Here directions are given and hence there is only one way. But if we are the ones who will be selecting the directions, then again there wil be infinite resolutions.hellraiser said:If we are given a vector and we find component of vector in given directions such that they are equivalent to the given vector, this is resoultion of vectors.P.
Normally if the angle between two vector is A, then the component of one in the direction of other is cosA times its magnitude. You can compare this with resolution along two axial planes. Let us consider a vector of magnitude x and making angle y with X-axis. Then it makes 90-y with Y-axis.hellraiser said:The problem I was doing was
Two vectors P and Q have resultant R. The resolved part of R in direction of P is Q.P.
If A be the angle between the vectors prove that
sin (A/2)=sqrt(P/2Q)
Vector components and resolved parts are two ways to break down a vector into its smaller parts or components. A vector is a mathematical quantity that has both magnitude (size) and direction. By breaking a vector into its components, we can better understand its properties and how it behaves in different situations.
To find the components of a vector, you can use trigonometric functions such as sine, cosine, and tangent. The magnitude of the vector can be multiplied by the cosine of the angle between the vector and the x-axis to find the x-component, and by the sine of the angle to find the y-component. Alternatively, you can use the Pythagorean theorem to calculate the components.
The main difference between vector components and resolved parts is the approach used to break down the vector. Vector components are found using trigonometric functions, while resolved parts are found using the Pythagorean theorem. They both provide the same information about the vector, but the methods used to find them may vary.
Understanding vector components and resolved parts is important in many scientific fields, including physics, engineering, and mathematics. By breaking down a vector, we can better understand its behavior and make accurate predictions about its movement and interactions with other vectors. This knowledge is crucial in solving real-world problems and developing new technologies.
Vector components and resolved parts are used in many real-life applications, such as navigation systems, robotics, and video games. For example, in a navigation system, vectors are used to represent the direction and distance between two locations. By breaking down these vectors into their components, the system can accurately calculate the best route to take. In robotics, understanding vector components and resolved parts is essential in programming movements and interactions between different parts of a robot. And in video games, vectors are used to simulate realistic movements and interactions between objects in the game world.