# vectors

A dot product, also called the scalar product is a special kind of product that takes coordinates of two vectors, and returns a number(scalar). The alternate name, ‘scalar product’ emphasises the fact that the result is a scalar.
Formula:

For 3-dimensions: $\overrightarrow{a}.\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3$

where, $\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\overrightarrow{b}=b_1\hat{i}+b_2\hat{j} + b_3\hat{k}$

Alternate formula for two vectors making an angle $\theta$: $\overrightarrow{a}.\overrightarrow{b}=|a||b|\cos{\theta}$
Physically, what one expects to get from a dot product is the product of a vector with that component of another vector which is along the first vector’s direction.
Some of the examples include, Work done, Electrical/Magnetic Flux, etc.

In the calculation of work, what one needs is the product of the Force and the Displacement in the direction of the force. So what we do is, we take the dot product of the Force vector, and the Displacement vector, which gives us the product of Force with the component of Displacement along the direction of the Force.

Mathematically, the formula for the dot product is simply the product of the magnitude of one vector with the projection of another vector on the first vector.
Projection(component) of a vector $\overrightarrow{a}$ along the direction of vector $\overrightarrow{b}$ is: $a_b(scalar)= |a|\cos{\theta}$ where $\theta$ is the angle between the two vectors.

NOTE: I have written the above answer for only 3 dimensional(geometrical) dot product.
The definition of the dot product can be extended for an n-dimensional space.
Another name for it is the inner product.
Hope you find it useful!!!