I'm reading up on dot products and keep seeing two different examples. One states that u[itex]\cdot[/itex]v = u[itex]_{i}[/itex][itex]\cdot[/itex]v[itex]_{i}[/itex] + u[itex]_{j}[/itex][itex]\cdot[/itex]v[itex]_{j}[/itex] Another: u[itex]\cdot[/itex]v = |u|[itex]\cdot[/itex]|v|cosθ I'm not understanding when to use the first or second method?
At the risk of stating something obvious, it depends on what information you are given and what you are trying to find. If you know the two vectors then you can find the dot product using the first equation. Then you can find the angle between the vectors using the second equation. On the other hand, if you are given the lengths of the vectors and the angle between them, you can use the second equation to find the dot product.
For example, if you are given that one vector is <1, 0, 0> and the other is <2, 2, 0> it is easy to calculate that the dot product is 1(2)+ 0(2)+ 0(0)= 2. But if you are given that one angle has length 1, the other has length [itex]2\sqrt{2}[/itex], and the angle between them is [itex]\pi/4[/itex], it is easiest to calculate [itex](1)(2\sqrt{2})(cos(\pi'4)= 2[/itex]. By the way, in spaces of dimension higher than 3, we can use the "sum of products of corresponding components" to find the dot product between two vectors, then use [itex]|u||v|cos(\theta)[/itex] to define the "angle between to vectors".