My situation: I am currently taking a course in linear algebra and I am quite all the time. I have however miraculously reached projections in 2-dimensions. I am studying and learning both from course material but also from this article and I am stuck about between page 6 and 7. My question is:
My problem: I wanted to know how to derive the formula for projection of a vector $a$ onto a linearly independent vector $b$, in order to understand because I am so lost. Why is this text, linked above, and my course book defining the dot product between two vectors as:
$$a\cdot b=|a|*|b|*cos(\theta)$$
and then calculating the length of the projection of the vector $a$ onto the vector $b$ as:
$$|a_b|=\frac{a\cdot b}{|a|}=\frac{a_x*b_x+a_y*b_y}{|a|}$$
However the dot product is defined as being the product of the length of $|a|$, $|b|$ and $cos(\theta)$ where $\theta$ is the angle between the two vectors. If I follow my course book, the slideshows provided by my professor and the pdf link above I am not in fact calculating the dot product and using it to calculate the length of the projection. I am in the example in the pdf calculating some random sum of $a_x*b_x+a_y*b_y$ instead of $\sqrt{a_x^2+a_y^2}*\sqrt{b_x^2+b_y^2}*cos(\theta)$.
How do you derive $\sqrt{a_x^2+a_y^2}*\sqrt{b_x^2+b_y^2}*cos(\theta) = a_x*b_x+a_y*b_y$ and why this difficult change? And why is this equation shift not mentioned? Am I missing something?