Investigate the distance between points and lines or planes.

Work in groups of 3–4, writing your answers on a separate sheet of paper.

Consider a line and a point.

We wish to find the (minimum) distance between the point and the line. Let’s solve this problem in as many ways as possible.

Draw the line segment from the point to the line of minimum length. Compare your drawing with others, make sure you all agree.
Solve this problem as you might in Calculus I: Find an explicit formula for a line, \(y = mx+b\), then minimize the distance (or the distance squared!) between the point and the points on the line.
Use facts about vectors and projection, \(\proj _{\vec {w}}(\vec {v})\), to find the minimum distance between the point and the line.
Use facts about the cross product to find the minimum distance between the point and the line.
Let:
\[ S(x,y) = (x-a)^2 + (y-b)^2 \]
for appropriate values of \(a\) and \(b\), and let \(\vecl (t)\) be a vector-valued function that parameterizes the line. Use \(S\) and \(\vecl \) to minimize the distance between the point and points on the line.
Let:
\[ S(x,y) = (x-a)^2 + (y-b)^2 \]
for appropriate values of \(a\) and \(b\). Use Lagrange multipliers to to find the minimum distance between the point and the line.
Now consider the point \((1,2,3)\) and the plane
\[ x + y+z = 0. \]
Use the method of your choosing to find the minimum distance between the point and the plane. Share your method with a friend.
Now consider the point \((1,2,3)\) and the sphere
\[ x^2 + y^2+z^2 = 1. \]
Use the method of your choosing find the minimum distance between the point and the sphere. Share your method with a friend.