Let's give this problem a try. So in this problem, we're told if vector v has an initial point (1,2) and a terminal point (4,4), sketch v as 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 x₂, this is going to be the x value for the terminal point, which is 4. So what I can first do is plug in 4 into this equation. 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 y₂. And y₂ is going to be this y value for the terminal point, which is 4, and then this is going to be minus the initial y, and the initial y is 2. So we're going to have 4 minus 2. So this right here gives us the vector (3,2).

Now that we have our vector in component form, what we can do is now sketch the position vector. 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 (3,2). So that means on the x-axis, we're going to move over 3 units, then on the y-axis, we're going to move up 2 units. This is what our vector is going to look like, and that's vector v sketched as a position vector.

We've now found the component form and have sketched our position vector. Our last step is going to be to calculate the magnitude of this vector. And to calculate the magnitude, we can use this equation: Magnitude = Vx2 + Vy2 , 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 32 plus 22. Now 32 is 9, so we're going to have the square root of 9+4 which 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 I hope you found this video helpful. Thanks for watching.