I have the equation of a line in the form $y = mx + b$. I also have a vector $a$.
How would I find the projection of $a$ onto the the line?
[Math] How to find the projection of a vector onto a line, given the equation of the line in the form $y = mx + b$
linear algebra
Best Answer
From the line equation, we find the slope to be:
$$\tan\alpha=\frac{dy}{dx}=m$$
So $$\sin\alpha=\frac{m}{\sqrt{m^2+1}},\cos\alpha=\frac{1}{\sqrt{m^2+1}}$$
The unit tangent vector $\vec{t}$ is given by:
$$\vec{t}=(\cos\alpha,sin\alpha)$$
The project of vector a onto this line is equal to $\vec{a}\cdot\vec{t}=a_x \cos\alpha +a_y \sin\alpha$.