(E4.1) In this problem, we explore how we can use what we know about vectors and the dot products to find the distance from a point to a line. Let L be the line with equation ~r(t) = 1 + t, 2 + 3t,1 t, and let Q = (1, 2, 3). a. Any point on L can be represented by (1 + t, 2 + 3t,1 t). Let P = (1 + t, 2 + 3t,1 t), find the vector P Q. b. Find the value of t such that P Q is perpendicular to the line. What are the coordinates for the point P? Draw a picture to illustrate the objects found so far. c. Find the distance between P and Q.

(E4.2) In this problem, we explore how we can use what we know about vectors and projections to find the distance from a point to a plane. Let be the plane with equation z = 4x + 3y + 4, and let Q = (4, 1, 8). a. Show that Q does not lie in the plane . b. Find a normal vector ~n to the plane . c. Find the coordinates of a point P in . d. Find the components of P Q. Draw a picture to illustrate the objects found so far. e. Explain why the magnitude of proj~n P Q gives the distance from the point Q to the plane f. Find this distance