Dot Product Projection: What Does A Dot B Mean?

In summary, the scalar projection of vector B onto vector A is given by B * Unit vector of A (or [A dot B]/[magnitude of A]). However, the dot product of simply A and B, assuming neither is a unit vector, does not have a clear geometric interpretation and its value can be used for various applications in mathematics and science such as calculating volumes.
  • #1
mvpshaq32
28
0
Simple question, but I don't know why I never learned this before.

If the scalar projection of vector B onto vector A is B * Unit vector of A (or [A dot B]/[magnitude of A]), then what does the dot product of simply A and B give you, assuming neither is a unit vector.

If it's not clear what I'm asking, it's that the component of vector B projected onto vector A is given by [A dot B]/[magnitude of A], but then what is the meaning of simply A dot B?
 
Last edited:
Mathematics news on Phys.org
  • #2
What do you mean with "meaning"?
Geometric interpretation? That is tricky.
Application in mathematics and science? It is very useful for many things.
 
  • #3
A.B = |A||B|cosx, where x is the angle between the vectors.
 
  • #4
mfb said:
What do you mean with "meaning"?
Geometric interpretation? That is tricky.
Application in mathematics and science? It is very useful for many things.

Yes, exactly, the geometric meaning.

What does its value represent?

For example the magnitude of the cross product represents the area of the parallelogram formed by two vectors.

So what does the dot product represent?
 
  • #5
You can use it to calculate volumes, for example, if you have the area of the floor given as (orthogonal) vector. This is used in the triple product.
 

1. What is the dot product projection?

The dot product projection is a mathematical operation used to calculate the scalar component of one vector in the direction of another vector. It is also known as the scalar projection or dot product.

2. How is the dot product projection calculated?

The dot product projection is calculated by multiplying the magnitude of one vector by the magnitude of the other vector, and then multiplying that product by the cosine of the angle between the two vectors.

3. What does the result of A·B mean?

The result of A·B represents the scalar component of vector A in the direction of vector B. It can also be interpreted as the magnitude of vector A multiplied by the magnitude of vector B, and then multiplied by the cosine of the angle between the two vectors.

4. What is the significance of the dot product projection in vector algebra?

The dot product projection is significant in vector algebra because it allows us to find the scalar component of one vector in the direction of another vector. This can be useful in various applications, such as calculating work done on an object or finding the angle between two vectors.

5. Can the dot product projection be negative?

Yes, the dot product projection can be negative. This occurs when the angle between the two vectors is greater than 90 degrees, resulting in a negative value for the cosine term in the calculation. A negative dot product projection indicates that the two vectors are pointing in opposite directions.

Similar threads

I Dot product, inner product, and projections
    • General Math
Replies
7
Views
894
MHB Find Dot Product Between Vector CD & Vector K
    • General Math
Replies
4
Views
3K
B Why is sine not used for dot product?
    • General Math
Replies
4
Views
2K
I Define Dot Product: A.B = ||A|| ||B|| cos(theta)
    • General Math
Replies
4
Views
2K
B Confused about dot product multiplication
    • Linear and Abstract Algebra
Replies
33
Views
852
I Inner product vs dot/scalar product
    • Calculus
Replies
4
Views
529
I Why is A.A = ||A||^2 in Vector Analysis?
    • General Math
Replies
7
Views
2K
Given value of vectors a,b, b.c and a+(b×c), Find (c.a)
    • Precalculus Mathematics Homework Help
Replies
5
Views
582
B Dot Products -- Some questions to help my understanding
    • Classical Physics
Replies
5
Views
1K
I Dot product of two vector operators in unusual coordinates
    • Quantum Physics
Replies
14
Views
1K

Hot Threads

Recent Insights

Back
Top