>> Okay, now let's do some calculations with the dot product just to review. If V has the components V1, V2, V3, and W has components W1, W2, W3, then to calculate V dot W in terms of their components, that's V1W1 plus V2W2 plus V3W3. In other words, we just take their same components, multiply their same components, coordinate wise, and then add the results. This formula, which appears in and of itself to be rather meaningless, actually has all those geometric properties that we just discussed. And in fact, it's easy to see if you look at those pictures, that this formula is a consequence. But in any case, let's use this formula to calculate V dot W, V dot V, length of V squared and the projection of W on V as well as the component of W perpendicular to V if we specifically have V as the vector with components 2, negative 2, 3 and W is equal to vector with components 5, 4, negative 2. So all we have to do is calculate. So for instance, to begin, V dot W. We multiply corresponding components and then add the results. 2 times 5, is 10. Negative 2 times 4, is negative 8. 3 times negative 2, is negative 6. And we wanna add these results. And so, 10 minus 8 is negative 2, excuse me, 10 minus 8 is plus 2, minus 6 is negative 4. What's the significance of the negative sign there? Remember, when the vectors make an angle greater than 90 degrees, then their dot product will be negative. How about V dot V? Well, V dot V is in effect the sum of squares of components 2 times 2 plus negative 2 times negative 2 plus 3 times 3, 4 plus 4 plus 9, 8 and 9 is 17. Now what about the length of V squared? Well remember, the squared length of any vector is just its dot product with itself. And so, we get 17. Now particular, that tells us that the length of V itself is the square root of 17.