What Are Dot Products and Vector Cross Products in Mathematics?

[woodie37]

Can someone explain to me what a dot product and vector product of two vectors are? Be as detailed as possible please! And also why does the right hand rule for vectors work?

 

[SystemTheory]

Cross Product Wiki:

http://en.wikipedia.org/wiki/Cross_product

Dot Product Wiki:

http://en.wikipedia.org/wiki/Dot_product

 

[charng]

Sure, I'd be happy to explain the dot product and vector product of two vectors, as well as the right hand rule for vectors.

First, let's start with the dot product. The dot product, also known as the scalar product, is a mathematical operation that takes two vectors and produces a scalar (a single number) as a result. It is denoted by a dot (·) between the two vectors. The dot product is calculated by multiplying the corresponding components of the two vectors and then adding them together.

For example, let's say we have two vectors, A = (3, 4) and B = (5, 2). The dot product of these two vectors would be A · B = (3*5) + (4*2) = 15 + 8 = 23. As you can see, the dot product of two vectors always results in a single number.

Now, let's move on to the vector product, also known as the cross product. Unlike the dot product, the vector product results in a vector as its output. It is denoted by a cross (×) between the two vectors. The vector product is calculated by taking the cross product of the two vectors and then finding the magnitude and direction of the resulting vector.

To find the cross product, we use the right hand rule. This rule states that if you point your right hand's index finger in the direction of the first vector, and your middle finger in the direction of the second vector, then your thumb will point in the direction of the resulting vector. This is why it is called the right hand rule.

The magnitude of the resulting vector is calculated by taking the product of the magnitudes of the two vectors and the sine of the angle between them. The direction of the resulting vector is perpendicular to both of the original vectors.

For example, let's say we have two vectors, A = (3, 4) and B = (5, 2). The cross product of these two vectors would be A × B = (3*2 - 4*5) = -14. The magnitude of this vector would be |A × B| = |-14| = 14. And using the right hand rule, we can determine that the direction of this vector is pointing out of the page.

So why does the right hand rule work? It is based on the fact that the cross product is a result of the vector components being