What is the difference between dot product and cross product in physics?

In summary, the dot product and cross product are two different ways to multiply vectors, and the formulas for each have been chosen to make them useful for different applications. The dot product is used for finding the angle between two vectors, and the formula θ = cos^-1[(A dot B)/|A||B|] can be rearranged from the dot product formula.
  • #1
physics kiddy
135
1
Why is dot product given by a*b cosθ whereas cross product ab sinθ.
 
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  • #2
Can you elaborate? Neither the dot product nor the cross product are "given" by either of those expressions.
 
  • #3
The cross product is a vector quantity so only the magnitude is given by a*b*sin θ. That does give you the dot product, but what exactly do you mean by "why"? Are you looking for a rationale for why mathematicians created such a thing? There are a lot of operations in math which were created because they are useful. And sometimes they are just created just purely as an intellectual curiosity.
 
  • #4
Dot product is a linear map. It's really a whole lot more complicated once you look at how dual vector spaces work, but let's forget that for a moment. Focus on linearity. You want property that a.b+a.c = a.(b+c) That tells you right away that a.(2b) = 2 a.b In other words, the result must be proportional to magnitude of b. Similarly, you can show that it's proportional to a. So you know the format must be ||a|| ||b|| * something. That something can only depend on directions of the vectors.

Next, if you dig a bit deeper into linear algebra, you'll see that (Ua).(Ub) = a.b, where U is any unitary transform. That is, if for example, you rotate both a and b vectors, their dot product doesn't change. That tells you that the dot product can only depend on the angle between the two vectors, not individual orientations. So you know that a.b = ||a|| ||b|| f(θ).

Showing that f(θ)=cos(θ) requires one more constraint. We want to make sure that length of our vectors is defined by the dot product itself. That is ||a||²=a.a That requires a function that satisfies f(0)=1, and then with some more hairy math you can get it to cos(θ). Unfortunately, I don't know a simple way to prove it that doesn't involve complex numbers.
 
  • #5
OK leave it. Please explain why do we use the formula,

θ = cos^-1[vector a * vector b/ a * b]

to find the angle between two vectors a and b.
 
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  • #6
Physics Kiddy
I see that you are still in high school. It's good that you are showing an interest is this so soon.
We can add, subtract and multiply ordinary numbers, and each of these operations are unique. There is only one way to multiply two numbers.

With vectors it is a little different. There is still only one way to add and subtract vectors (head to tail scheme you may be familiar with) but there are several different ways to multiply vectors.

In both cases we are multiplying their magnitudes (this makes sense right).
Here is the rationale for the difference:

First dot product:
If the vectors are pointing in the same direction the dot product gives you the full product of their magnitudes |A||B| since cos(0) =1. If the vectors are not pointing in the same direction the dot product gives you less. When they are pointing at 90 degree angles to each other it gives you zero. In other words the dot product is the actual product but reduced as the angle between the vectors increases. Imagine pulling a wagon by its handle. You are trying to pull it horizontally, but the handle is at some angle. The steeper the angle the less of your pulling effort works horizontally. If the angle is at 90 degrees (straight up) you will not be able to pull the wagon horizontally no matter how hard you pull. You will be able to pull it the easiest if the handle is horizontal (angle 0). This is an example of use of dot product.

Cross product:
Here if the vectors are at 90 degree angles to each other you get the full product of their magnitudes |A||B|. As the angle *decreases* you get less, and at 0 degrees you get 0 (just opposite of dot product). Imagine pulling on a big wrench in order to twist off a tight bolt. Your best bet is to pull in a direction that is at a right angle to the wrench. If you pull at a smaller angle, less of your effort will be torquing the bolt. If you pull at a 0 degree angle (along the length of the wrench) none of your effort is working to untwist bolt. This is an application where cross product is used.

You can rearrange dot product formula from:
A dot B = |A||B|cosθ
to
θ = cos^-1[(A dot B)/|A||B|]
Which is convenient for finding angle given two vectors.
 
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1. What is the difference between the cross product and dot product?

The cross product and dot product are both mathematical operations used in vector algebra. The main difference between them is that the cross product results in a vector, while the dot product results in a scalar. This means that the cross product takes two vectors and produces a third vector that is perpendicular to both, while the dot product takes two vectors and produces a scalar value that represents the magnitude of the projection of one vector onto the other.

2. When should I use the cross product and when should I use the dot product?

The cross product is useful for determining the direction of a vector perpendicular to two other vectors, which is important in many physical applications such as calculating torque or magnetic fields. The dot product, on the other hand, is useful for calculating the angle between two vectors or determining the component of one vector in the direction of another. It is also used in many mathematical and engineering calculations.

3. How do I calculate the cross product and dot product?

The cross product is calculated by taking the determinant of a 3x3 matrix composed of the two vectors and their unit vectors. The dot product is calculated by multiplying the corresponding components of the two vectors and then adding them together. Both operations have specific formulas and can also be calculated using geometric methods.

4. Are there any properties of the cross product and dot product that are important to know?

Yes, there are several properties of these operations that are important to know. The cross product is anti-commutative, meaning that changing the order of the vectors will result in a vector in the opposite direction. It is also distributive and follows the right-hand rule. The dot product is commutative, meaning that changing the order of the vectors does not change the result. It is also distributive and follows the law of cosines.

5. Can the cross product and dot product be used in higher dimensions?

Yes, both the cross product and dot product can be used in higher dimensions, but they are not as commonly used as in three dimensions. In higher dimensions, the cross product involves taking the determinants of larger matrices and the dot product involves multiplying more components. In some cases, alternative operations may be used instead, such as the wedge product for the cross product in four dimensions.

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