If vectors v⃗=⟨4,1⟩v ⃗=⟨4,1⟩v⃗=⟨4,1⟩, u⃗=⟨−8,3⟩u ⃗=⟨-8,3⟩u⃗=⟨−8,3⟩, and w⃗=⟨−2,−1⟩w ⃗=⟨-2,-1⟩w⃗=⟨−2,−1⟩, calculate w⃗−3(v⃗+u⃗)w ⃗-3(v ⃗+u ⃗)w⃗−3(v⃗+u⃗).

⟨ 10 , − 13 ⟩ \langle10,-13\rangle ⟨ 10 , − 13 ⟩

If vectors v ⃗ = ⟨ 4 , 1 ⟩ v ⃗=⟨4,1⟩ v ⃗ = ⟨ 4 , 1 ⟩ , u ⃗ = ⟨ − 8 , 3 ⟩ u ⃗=⟨-8,3⟩ u ⃗ = ⟨ − 8 , 3 ⟩ , and w ⃗ = ⟨ − 2 , − 1 ⟩ w ⃗=⟨-2,-1⟩ w ⃗ = ⟨ − 2 , − 1 ⟩ , calculate w ⃗ − 3 ( v ⃗ + u ⃗ ) w ⃗-3(v ⃗+u ⃗) w ⃗ − 3 ( v ⃗ + u ⃗ ) .

Let's give this problem a try. So here we have 3 vectors, vector v, which is 41, vector u, which is negative 83, and vector w, which is negative 2 negative 1. And we're asked to figure out this entire operation, which is w minus 3 times v plus u. Now, whenever I'm starting these types of problems where we have a lot of vector operations happening, I like to start small with the problem and then work my way to figuring out the whole thing. So to start things off, I'm just gonna figure out what 'v plus u' is. Now to find 'v plus u', I can just add the individual vectors. So you can see that vector v is 41, and then I can see here that vector u is negative 83. And this operation says that we're going to add these vectors together. Now, adding these vectors I can add the individual components. I can add the 4 and the negative 8, which is going to give me negative 4, and I can add the 1 and the 3, which is going to give me positive 4. So this is the vector v plus u. Now from here what I can do is I can incorporate this 3, because notice we have a 3 being multiplied on the outside of v plus u. So to find 3 times 'v+u', well, we already figured out that 'v+u' is this vector right here. So I just need to multiply this vector by 3. So we're gonna have 3 times negative four, four. Now recall when multiplying scalars by vectors, you can distribute the scalar to each component. So we're gonna have 3 times negative 4 which is negative 12, and then 3 times positive 4 which is positive 12, and that's going to be our vector 3 times v plus u. Now our last step here is going to be incorporating the w, and I can see that we have a w here and we're subtracting off this entire vector. So to find w minus 3 times v plus u, we're going to take our vector w, which I can see here is negative 2 negative 1, and then we're going to subtract off the vector 3 times v plus u, which is the vector we just figured out up here. So subtract off negative 12, 12, and to do this well, I just need to subtract the individual components. So we're going to have negative 2 minus negative 12, and Whenever you have minus a negative that's the same thing as plus. So this is the same thing as negative 2 plus 12, and negative 2 plus 12 should give you positive 10. Then we're going to take a negative one, and we're going to subtract from negative one positive 12. And negative one minus 12, well, if you have something if you have if you owe $1 and then you owe another $12 you owe $13 so that means we have negative 13. So this right here is gonna be the vector w minus 3 times v plus u. And notice how when taking this by the steps, we were able to figure out this whole operation up here. So this would be the solution to our problem. So hope you found this video helpful. Thanks for watching.