The point [math]R[/math] is any point on the line (this changes as we vary [math]\lambda[/math])[br]The vector [math]PR[/math] is the vector from the point to the line. In this case, we want to find the point [math]R[/math] such that [math]|PR|=5\sqrt{3}[/math]

- Define [math]R[/math] as a general point on the line[br]- Find the vector [math]PR[/math][br]- Set [math]|PR|=[/math]length, solve for [math]\lambda[/math].[br]Note that there will generally be 2 solutions, unless the line is tangent to the sphere (min distance = length), or does not intersect (min distance < length)

Moves the point [math]R[/math] along the line until it is at the correct distance (indicated by the sphere)

Considering the minimum distance from the point [math]P[/math] to the line. There will generally be 2 solutions, unless the line is tangent to the sphere (min distance = [math]d[/math]), or does not intersect (min distance < [math]d[/math])

- Define [math]R[/math] as a general point on the line[br]- Find the vector [math]PR[/math][br]- Find an expression for [math]|PR|[/math], find the value of [math]\lambda[/math] such that this is minimised.[br](This could also be done by considering when [math]PR[/math] is perpendicular to the line, i.e. when [math]PR\cdot u=0[/math] and finding [math]\lambda[/math] that way.