How Do Dot Products Reflect Vector Projections?

[Pochen Liu]

I know that a dot product of 2, 2 dimension vectors a, b =

(a_x * b_x) + (a_y * b_y)

but it also is equal to

a*bCos(θ)

because of "projections". That we are multiplying a vector by the 'scalar' property of the other vector which confuses me because that projection is in the direction of the other vector, therefore no scalar, so I cannot see how these two produce the same outcome.

What am I missing intuitively?

 

[jedishrfu]

Better Explained might help you with the concept

https://betterexplained.com/articles/vector-calculus-understanding-the-dot-product/

I look at as a projection of one vector on another ie what is As projection on B and similarly what is Bs projection on A.