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Line integrals in vector fields articles Video transcript One of the most fundamental ideas in all of physics is the idea of work. Now when you first learn work, you just say, oh, that's just force times distance. But then later on, when you learn a little bit about vectors, you realize that the force isn't always going in the same direction as your displacement.
So you learn that work is really the magnitude, let me write this down, the magnitude of the force, in the direction, or the component of the force in the direction of displacement. Displacement is just distance with some direction. Times the magnitude of the displacement, or you could say, times the distance displaced.
And the classic example.
Maybe you have an ice cube, or some type of block. I just ice so that there's not a lot of friction. Maybe it's standing on a bigger lake or ice or something.
And maybe you're pulling on that ice cube at an angle. Let's say, you're pulling at an angle like that. That is my force, right there. Let's say my force is equal to-- well, that's my force vector. Let's say the magnitude of my force vector, let's say it's 10 newtons.
And let's say the direction of my force vector, right, any vector has to have a magnitude and a direction, and the direction, let's say it has a 30 degree angle, let's say a 60 degree angle, above horizontal. So that's the direction I'm pulling in. And let's say I displace it.
This is all review, hopefully.
If you're displacing it, let's say you displace it 5 newtons. So let's say the displacement, that's the displacement vector right there, and the magnitude of it is equal to 5 meters.
So you've learned from the definition of work, you can't just say, oh, I'm pulling with 10 newtons of force and I'm moving it 5 meters.
You can't just multiply the 10 newtons times the 5 meters. You have to find the magnitude of the component going in the same direction as my displacement.- Elementary Arithmetic - High School Math - College Algebra - Trigonometry - Geometry - Calculus But let's start at the beginning and work our way up through the various areas of math.
We need a good foundation of each area to build upon for the next level. In Section VO we defined vector addition and scalar multiplication.
These two operations combine nicely to give us a construction known as a linear combination, a construct that we .
Browse other questions tagged linear-algebra matrices or ask your own question. asked. 5 years, 7 months ago. viewed. 57, times. active. 1 year ago. 29 votes Writing a vector as a linear combination of other vectors.
1. linear combination of the eigenvectors of matrix M. 4.
A linear combination of two or more vectors is the vector obtained by adding two or more vectors (with different directions) which are multiplied by scalar values. Examples Write the vector = (1, 2, 3) as a linear combination of the vectors: = (1, 0, 1), = (1, 1, 0) and = (0, 1, 1).
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Using line integrals to find the work done on a particle moving through a vector field.
Linear Combinations of Vectors – The Basics In linear algebra, we define the concept of linear combinations in terms of vectors. But, it is actually possible to talk about linear combinations of anything as long as you understand the main idea of a linear combination. algorithm. A series of repeatable steps for carrying out a certain type of task with data. As with data structures, people studying computer science learn about . Writing a Vector as a Linear Combination of Other Vectors Sometimes you might be asked to write a vector as a linear combination of other vectors. This requires .