I want to know the points of a vector when I know one of the points and also the distance between the two.

Question

So I have a vector equation which is

$$(1,2,-1) + t(2,1,-2)$$

I also know a point of that line which is

$$(0,0,1)$$

I need to know all the points in which the distance is 3 from the point at (0,0,1)

Thank you.

Answer 1

It's a bit unclear, but I think you are looking for the values of $t$ such that the distance from (0,0,1) to (1,2,-1)+t(2,1,-2) is 3. The Euclidean distance between the point and a parameterized point on the line can be written as $$ \sqrt{(1+2t-0)^2+(2+1t-0)^2+(-1-2t-1)^2} $$ so you are looking for $t$ such that $$ \sqrt{(1+2t)^2+(2+t)^2+(-2-2t)^2}=3 $$