Dot product (scalar product) of 2 vectors: ABcos[itex]\theta[/itex]

In summary, the conversation is discussing the derivation of the Bcosθ term and how it fits into the equation A x B = [ABcosθ]. The person is confused about where the A variable comes from and how Bcosθ is derived. They share a link for further understanding and mention that they will need to do more self-study before coming back with a more specific question. Eventually, they figure out the concept and plan to start a new thread for another vector question.
  • #1
LearninDaMath
295
0
scalarproduct.png




How, precisely, do you get/derive the Bcosθ term?



Is it simply [Cosθ=A/B] --> [BCosθ = A] ? It can't be that simple because then how is the extra length of vector A fit into [*A*Bcosθ]? I feel pretty confused as to what is going on here. To summerize, A x B = [ABcosθ] makes little sense.. and i think that is because for one reason, I'm not sure how Bcosθ is derived. And the other reason is because I don't know where they get the A variable either. Just can't seem to see what's going on here.
 
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  • #3


elfmotat said:

Wow, that looks a little more complex than I thought it would be. I thought it was something much more straight forward. I'll have to do some further self study for a little while and get back to you on what I was able to figure out. At worst, I hope to at least come back with a more specific question. At best, I'll be able to figure it out. Thanks.
 
  • #4
Figured it out, this makes perfect sense now. However I have another vector question, but unrelated to this one, so I'll start a new thread for it.
 
  • #5


The dot product, also known as the scalar product, is a mathematical operation that takes two vectors and produces a scalar quantity. In the case of two vectors, A and B, the dot product is defined as ABcosθ, where θ is the angle between the two vectors. The question of how the term Bcosθ is derived is a common one, and it is important to understand the concept behind it.

To derive the Bcosθ term, we must first understand the geometric interpretation of the dot product. The dot product of two vectors, A and B, can be thought of as the projection of vector A onto vector B, multiplied by the length of vector B. This can be visualized as the shadow of vector A cast onto vector B, with the length of B representing the size of the screen onto which the shadow is cast.

Now, if we consider the angle θ between the two vectors, we can see that the projection of A onto B will be equal to the length of A multiplied by the cosine of θ. This is because the cosine of θ is equal to the adjacent side of the right triangle formed by A and B, divided by the hypotenuse (which is the length of B). This is where the Bcosθ term comes from in the dot product formula.

To further clarify, let's consider the dot product in terms of its components. If we have two vectors, A = (ax, ay, az) and B = (bx, by, bz), then the dot product AB can be written as:

AB = axbx + ayby + azbz

We can see that the Bcosθ term is represented by by, which is the y-component of vector B. This is consistent with our geometric interpretation, as the y-component of vector B represents the length of B multiplied by the cosine of θ (since the projection of A onto B is in the direction of the y-axis).

In summary, the Bcosθ term in the dot product formula is derived from the geometric interpretation of the dot product, where it represents the projection of vector A onto vector B, multiplied by the length of B. This term is essential in calculating the dot product and understanding its significance in vector operations.
 

What is the definition of dot product of two vectors?

The dot product, also known as the scalar product, is a mathematical operation that takes two vectors in a multi-dimensional space and produces a scalar quantity. It is calculated by multiplying the magnitudes of the two vectors and the cosine of the angle between them.

How do you calculate the dot product of two vectors?

The dot product of two vectors A and B can be calculated by taking the product of their magnitudes (|A| and |B|) and the cosine of the angle between them (θ). This can be represented by the formula: A · B = |A||B|cosθ.

What is the significance of the dot product in vector operations?

The dot product has many applications in vector operations, including finding the angle between two vectors, determining if two vectors are perpendicular, calculating the work done by a force, and finding the projection of one vector onto another.

What are some properties of the dot product?

The dot product has several important properties, including commutativity (A · B = B · A), distributivity (A · (B + C) = A · B + A · C), and associativity (k(A · B) = (kA) · B = A · (kB)), where k is a scalar. It also follows the distributive law and the laws of exponents.

Can the dot product be negative?

Yes, the dot product can be negative if the angle between the two vectors is obtuse (greater than 90 degrees). This means that the two vectors are pointing in opposite directions, resulting in a negative product. If the angle is acute (less than 90 degrees), the dot product will be positive.

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