14. Vectors
Vectors in Component Form
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Let's give this problem a try. So in this problem, we're told if vector v has initial point 12 and terminal point 44, Sketch v is a position vector and calculate its magnitude. Now, to solve this problem, the first thing I'm going to do is figure out what vector v is in component form. This is always a good first step when you're dealing with these types of problems where you're given the initial and terminal point of a vector. Now to solve this, to find this component form, what I can do is take the final x and subtract off the initial x, and then take the final y and subtract off the initial y, and that's going to give us our vector. Now, first off, we can see that for x 2, this is gonna be the x value for the terminal point, which is 4. So what I can first do is plug in 4 to this equation. Now Now next, I'm going to subtract off the initial x, and the initial x is 1. So we're going to have 4 minus 1 for the difference in the x's. Now as for the y's, I can see that we have y 2. And y 2 is going to be this y value for the terminal point, which is 4, and then this is gonna be minus the initial y, and the initial y is 2. So we're going to have 4 minus 1, which is 3 comma 4 minus 2, which is 2. So So this right here is the vector. So now that we have our vector in component form, what we can do is we can now sketch the position vector. And the position vector just means that our vector is going to start at the origin. So if I go here at the origin of our graph, I can see that our vector is 3. So that means on the x axis, we're gonna go over here 1, 2, 3, then I can see that our y value is 2, so that means we're going to go up 2. So So this is what our vector is going to look like, and that's vector v sketched as a position vector. So we've now found the component form, and we've sketched our position vector. And our last step is going to be to calculate the magnitude of this vector. And to calculate the magnitude, we can use this equation. The magnitude of v is equal to the square root of vx squared plus vy squared, where vx is the x component and vy is the y component. So we can see that we have these values that we calculated already, so we're going to have the vx which is 3, so we have 3 squared plus 2 squared. Now 3 squared is 9, so we're gonna have the square root of 9 plus 2 squared which is 4. And 9+4 is 13. So the magnitude of our vector 'v' is equal to the square root of 13. And this is the magnitude of our vector, as well as our vector sketched as a position vector, and that is the solution to this problem. So hope you found this video helpful. Thanks for watching.
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