Why is the error vector $e = b - p$ ( Introduction to Linear Algebra of Gilbert Strang)

Question

I have a question in the projection parts (chapter 4 in his book). I don't really understand why $e=b-p$ (when $e$ is error vector). A line goes through the origin in the direction $a=(a_1,...,a_m)$. So we will project $b$ into that line. Can you guys help me to explain it?

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Answer 1

When we project $\mathbf{b}$ onto the vector $\mathbf{a}$, to say a point $p$ in $\mathbf{a}$, $\mathbf{e}=\mathbf{b}-p$ is a line from the point of projection $p$ to the vector $\mathbf{b}$.

The concept of error comes from the line fitting problem.

Finding the best least squares approximation of a line $L$ passing through 3 given vectors $(\mathbf{x_1}, \mathbf{x_2}, \mathbf{x_3})$ for instance can be thought of as minimising the square of projection distance of the vectors $(min(|\mathbf{x_1} - p_1|^2 + |\mathbf{x_2}-p_2|^2 + |\mathbf{x_3} - p_3|^2)$, where $(p_1, p_2, p_3)$ are the points of projection on the line L for the vectors $(\mathbf{x_1}, \mathbf{x_2}, \mathbf{x_3})$ respectively.