 We saw that the dot product of a vector with itself can be interpreted as the square of the magnitude of the vector. But does the dot product of two arbitrary vectors tell us anything useful? Maybe? But let's see if we can find some properties of the dot product first. One useful property occurs when we have three vectors, and let's find and compare the dot product of one vector with the sum and the sum of the dot products. Essentially we're looking at the distributive property. So to find the dot product of a vector with a sum of two vectors, parentheses mean their usual thing, so we'll take care of the operation inside the parentheses first and add the vectors v and w to get, and then we can take the dot product and find. Meanwhile, if we want to find the sum of the dot products, we'll find the dot products first, so we'll find u dot v and u dot w, and we can add them to get, and based on one example we can conclude that the dot product is distributive, assuming that both sides are defined, the dot product of a sum is the sum of the dot products. Now, basic things on one example or even no examples at all and just something you made up on the spot is good enough for politicians and political pundits, but we should try to do better. So we can prove this by generalizing an example. So here we took three very specific vectors, so let's use three very generic vectors, and so we'll try to do the same thing. We'll find the dot product of the sum, so we'll add v and w, then take the dot product with u. Next, we'll find the sum of the dot products, so we'll find the individual dot products, and so the sum of the dot products will be, and these two are not quite the same, but we can do a little bit of algebra. Now, since factored form is best, let's see if we can do any useful factorizations on this second expression, and we have a guide for where we want to end up. Notice that our first expression is u1 times something plus u2 times something plus u3 times something. So let's see if we can rewrite our second expression as u1 times something plus u2 times something plus u3 times something, and we see that we do have a u1 times v1 plus w1, and similarly we can factor out a u2 and a u3, and we can see that these are the same thing and consequently we can conclude that the dot product of a sum is the sum of the dot products. Although we should add a disclaimer, because we used vectors in R3, our result is only proven for vectors in R3. Well, let's try another experiment. Let's say we take two vectors, does the order of the dot product matter? So we find u.v and v.u, and again generalizing from one example, we conclude that the dot product is commutative. u.v is the same as v.u. We should prove this, and we can prove this by generalizing an example. The only problem is we have to figure out where our example will come from, so let's suppose we have two vectors in R4. Since u and v are vectors in R4, they have four components, and so we'll let them be. And so we find their dot products, and oh my, look at the time.