Find the work done by a force $ F = 8i - 6j + 9k $ that moves an object from the point $ (0, 10, 8) $ to the point $ (6, 12, 20) $ along a straight line. The distance is measured in meters and the force in newtons.
So if you want to find how much work is done, uh, first we'll need to know our force and our distance vectors. And then we are displacement vectors, and then we just take the dot product of those. So first, I'm going to just rewrite our force in terms of a broker in the bracket notation to just be eight negative 69 because I just like that better than the I J k notation. Um, no real reason. Well, I'm doing it. So, uh, and then to figure out our displacement, we're going to do our end minus r. Starts risking Subtract these two factors here, give us d is equal to So remember we subtract. Component was six minus zero is 6 12 minus 10 is 2 20 minus eight is 12. Now we just take the dot product of both of those. So work is going to be 6 to 12 dotted with eight negative 69 And even though I did displacement dotted with force, it doesn't really matter because their communicative um but what does matter is what I do. These remember, I multiply each of them component twice and then add it all up. So this is going to give us 48 minus 12 and then plus one a week. And if we were to add all these together, um, so 108 minus 12 plus 48 that gives 144 and the units with these would be, uh, Newton metres, which is the same thing as jewels, would be 140 for jewels for the units.