Apologies if this is extremely simple. I'm a biology undergrad and I need this to measure cell membranes but I'm really struggling to find a solution anywhere.
I'm trying to find the gradient of a tangent line given a tangent vector $(a, b)$ so that I can plot the tangent line. I think I therefore have the $x$ and $y$ components of a velocity vector. How would I find the gradient of the velocity vector to plot this in the form $y=mx+c$ ?
If the tangent vector is $(a,b)$ where $a$ is the $x$ component and $b$ is the $y$ component, that means the vector "rises" $b$ units in the $y$ direction while "running" $a$ units in the $x$ direction. (Negative numbers reverse the direction, so for example $b=-2$ means you fall $2$ units and $a=-3$ means you travel $3$ units to the left.)
The ratio of rise/run is exactly what $m$ represents in $y=mx+c.$ So you want something of the form
$$ y = \frac ba x + c.$$
You now need to set $c$ so that your line passes through the tangent point. Set $x$ and $y$ to their values at the tangent point, solve the equation for $c,$ and that will be the value of $c$ you want.