Shedding some light on the dot product

In summary: V.W/W)*W.In summary, the dot product A . B is the magnitude of vector A times the projection of B onto A, and B . A is the magnitude of vector B times the projection of A onto B. It is commutative and can be used to find the components of a vector in the direction of another vector by multiplying the vector by the unit vector of the direction and the magnitude of the projection.
  • #1
cytochrome
166
3
The dot product A . B is the magnitude of vector A times the projection of B onto A.

B . A is the magnitude of vector B times the projection of A onto B.

Correct?

A . B = B . A and this makes sense. But, say you're trying to find the components of a vector V in the direction of a vector W. Would it matter whether or not you wrote V . W or W . V?

EDIT: Also, does anyone know what it means (geometrically speaking) to find the components of a vector in the direction of another vector? I can give an example from a book if needed.
 
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  • #2
To the first question, no it wouldn't matter. The dot product is commutative so ##\vec{V} \cdot \vec{W} = \vec{W} \cdot \vec{V}## for any vectors V and W.

To the edit, imagine you have your vector lying in the plane. Now imagine it is the hypotenuse of a right triangle where one of the sides of the triangle is parallel to the x-axis and the other side is parallel to the y-axis. The component of the main vector (which remember is represented as the hypotenuse) in the x direction is the length of the side of the triangle parallel to the x-axis, and the same for the y direction.
 
  • #3
cytochrome said:
The dot product A . B is the magnitude of
A . B = B . A and this makes sense. But, say you're trying to find the components of a vector V in the direction of a vector W. Would it matter whether or not you wrote V . W or W . V?

It does not matter which way you write. But none of them will give you the component of V along the direction of W.
You need to multiply V by the unit vector along W.
So the magnitude of the projection is given by V.W/W
 

Related to Shedding some light on the dot product

1. What is the dot product?

The dot product is a mathematical operation that takes two vectors as input and returns a single scalar value. It is also known as the inner product or scalar product.

2. How is the dot product calculated?

The dot product is calculated by multiplying the corresponding elements of the two vectors and then summing the results. For example, if we have two vectors, A = [a1, a2, a3] and B = [b1, b2, b3], then the dot product would be calculated as a1*b1 + a2*b2 + a3*b3.

3. What is the significance of the dot product?

The dot product has many applications in mathematics, physics, and engineering. It can be used to calculate the angle between two vectors, determine the projection of one vector onto another, and solve systems of linear equations.

4. Can the dot product be negative?

Yes, the dot product can be negative. This occurs when the angle between the two vectors is greater than 90 degrees. In this case, the dot product represents the negative of the magnitude of the projection of one vector onto the other.

5. Is the dot product commutative?

No, the dot product is not commutative. This means that the order in which the vectors are multiplied matters. In other words, A · B is not equal to B · A. However, it is associative, meaning that (A · B) · C = A · (B · C).

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