Finding distance from point to line determined by vector

Question

We are given a point, b. We are given a vector, v, which determines a line. We are given a point c such that there exists a line through c which has the same direction as the line determined by v. (Obviously we also know a vector, c, given this point).

How can we find the distance from the point b to the line determined by c?

I know that we have to find a projection and use that to find the orthogonal line from b to the line determined by c. However, I have no idea how to find the line determined by the point c and the direction of the vector v.

If the line determined by the vector v, and the new line through c, have the same direction, then they are parallel. However, I'm not sure how to calculate what this line actually is.

EDIT: Here's an example:

b = point (1, 2, 2, 4)
v = vector (1, 7, 7, 1)
c = point (4, 1, 2, 2)

I'm confident that I can work this out on my own, once I can figure out how to determine the line through c with direction given by the vector v.

Answer 1

the line is given by $$ l(t) = c + tv $$ the vector from the Point c on the line to b (not on the line) is $$(b-c) $$ now you can compute the projection of this vector onto the line by $$proj(b-c) = \frac{ (b-c) \cdot v }{ ||v||^2 }v$$ Now you can compute the rejection, the vector from the projection to b by $$ rej = (b-c) - proj(b-c) $$ it's norm is the distance you are looking for $$ dist = || rej || $$

Answer 2

Hint:

Maybe you want to find stationary points of $f(t)=| c+tv-b|^2$, $t$ a scalar.

Answer 3

Here is my try. You can set up a plane with vector $v$ as its normal, plane passing though point $b$. It would look like $x+7y+7z+w=33$ Now you can set up a 4dim line with $v$ as its directional vector and passing through point $c$, so that could look like $<4,1,2,2> + k<1,7,7,1>$ so that gives $x=4+k, y=1+7k, z=2+7k, w=2+k$ Substitute this in your plane equation and find the value of $k$. That gives you the point of intersection of the line with the plane. Calculate the distance between $b$ and the point of intersection with the distance formula.