Dot Product (Scalar Product)

The dot product is a mathematical operation that produces a scalar quantity (i.e., a quantity with magnitude only, no direction) by taking the product of the components of the input vectors and summing them. It is defined by the following formula:

$A\cdot B=A_1\ast B_1+A_2\ast B_2+...+A_n\ast B_n$

where A and B are the input vectors, and $A_1,A_2,...,A_n$ and $B_1,B_2,...,B_n$ are the components of the vectors.

Example

In our example, the dot product of vectors A and B is calculated as follows:

$A\cdot B=A_1\ast B_1+A_2\ast B_2$ $=3\ast 2+4\ast 1$ $=7$

Once we have calculated the dot product of the vectors, we can use it to calculate the angle between vectors using the following formula:

$\theta =\arccos \left(\frac{A\cdot B}{|A|\ast |B|}\right)$

where θ is the angle between the vectors, $A\cdot B$ is the dot product of the vectors, and $|A|$ and $|B|$ are the magnitudes of the vectors. In our example, we substitute the values we have calculated into this formula to get the following:

$\theta =\arccos \left(\frac{7}{5\ast 2.236}\right)$ $=\arccos \left(\frac{7}{11.18}\right)$ $=57.22$ degrees

This means that the angle between vectors A and B is 57.22 degrees.

#sapling