Dot Product: Understand When to Use Each Method

[mindheavy]

I'm reading up on dot products and keep seeing two different examples.

One states that u$\cdot$v = u$_{i}$$\cdot$v$_{i}$ + u$_{j}$$\cdot$v$_{j}$

Another: u$\cdot$v = |u|$\cdot$|v|cosθ

I'm not understanding when to use the first or second method?

 

[jbunniii]

I'm reading up on dot products and keep seeing two different examples.

One states that u$\cdot$v = u$_{i}$$\cdot$v$_{i}$ + u$_{j}$$\cdot$v$_{j}$

Another: u$\cdot$v = |u|$\cdot$|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.

 

[mindheavy]

Makes sense, I think the way the book I'm looking in words it was confusing me. Thanks

 

[HallsofIvy]

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 $2\sqrt{2}$, and the angle between them is $\pi/4$, it is easiest to calculate $(1)(2\sqrt{2})(cos(\pi'4)= 2$.

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 $|u||v|cos(\theta)$ to define the "angle between to vectors".

 

[CellCoree]

The first method, u\cdotv = u_{i}\cdotv_{i} + u_{j}\cdotv_{j}, is used to calculate the dot product of two vectors in Cartesian coordinates. This method is useful when working with vectors in 2D or 3D space, as it allows us to break down the vectors into their components and calculate the dot product for each component separately.

The second method, u\cdotv = |u|\cdot|v|cosθ, is used to calculate the dot product of two vectors in polar coordinates. This method is useful when working with vectors in a polar coordinate system, such as when dealing with circular motion or forces acting at an angle.

It is important to understand when to use each method depending on the context of the problem you are working on. If you are working with vectors in Cartesian coordinates, use the first method. If you are working with vectors in polar coordinates, use the second method. It is also important to note that both methods will give the same result for the dot product, so you can choose the method that is most convenient for you.