Q:

Show $ a·b = |a| × |b| \cos(\theta)$ geometrically and by using no algebraic arguments at all

I want to know if there is a more natural way of deriving $ a·b = |a| × |b| \cos(\theta)$ without using algebraic identities and looking at a figure instead. I am familiar with the algebraic method. 

A:

The dot product of two Euclidean vectors $a$ and $b$ is defined by the same formula you noted. Also, the scalar projection of a Euclidean vector $a$ onto a Euclidean vector $b$ is known by
$$a\cdot b=|a|\cos\theta$$ 
where $θ$ is the angle between two vectors $a$ and $b$.For having a geometric definition of the dot product, this can be rewritten as:
$$a_b=|a|\cdot\widehat{\mathbf b}$$
where $\widehat{\mathbf b} = b/| b|$ is the unit vector in the direction of B. So The dot product is thus characterized geometrically by
$$a\cdot b = a_b|b|$$.

