What point on the line y=3x-1 is closest to the origin? What is the distance?

[skyturnred]

Homework Statement

What point on the line y=3x-1 is closest to the origin? What is the distance to the origin from this point?

Homework Equations

The Attempt at a Solution

OK, I thought I knew how to do this but I don't think my answer seems right.

I find the normal vector to this line to be n=[3, -1]. I find an arbitrary point on the line by imputing x=1 into the equation and getting y=2 (so the point on the line is (1,2)). Then I subtract the vector of the origin (0,0) from that to get a random vector from the origin to a random point on the line. This vector is w=[1,2].

So to get the point on the line I do the projection of w onto n. My answer is [(3/11),(-1/11)]. This is the shortest vector from the origin to the line.. right? Can someone just double check this for me?

So then to get the point on the line.. I thought I should plug (3/11) into the original equation and get an answer for y but I get y=-2/11. Does this seem right?

 

[MathMarvel]

You can find the line that is both perpendicular to the given line and passes through the origin and see where these two lines intersect. This will be the point you're looking for.