1. Use normal vectors to determine the interaction, if any, for each of the following pairs of planes. Give a geometric interpretation in each case and the number of solutions for the corresponding system of linear equations. If the planes intersect in a line, determine a vector equation of the line.
   1. x + 2y + 3z = 4 2x + 4y + 6z = 10
   2. 2x y + 2z = 8 4x 2y + 4z = 16
   3. x + 3y 5z = 12 2x + 3y 4z = 6
2. a. Determine an equation of the line of intersection of the planes 4x 3y z = 1 and 2x + 4y + z = 5. b. Find the scalar equation for the plane through (4, 5, -3) and perpendicular to that line of intersection.
3. Use normal vectors to determine the intersection, if any, for each of the following groups of three planes. Give a geometric interpretation in each case and the number of solutions for the corresponding linear system of equations. If the planes intersect in a line, determine a vector equation of the line. If the planes intersect in a point, determine the coordinates of the point.
   1. x + 2y + 3z = 4 2x + 4y + 6z = 7 x + 3y + 2z = 3
   2. x + 2y + 3z = 4 2x + 4y + 6z = 7 3x + 6y + 9z = 5
   3. x + 2y + z = 2 2x + 4y + 2z = 4 3x + 6y + 3z = 6
   4. x 2y 2z = 6 2x 5y + 3z = 10 3x 4y + z = 1
   5. x y + 3z = 4 x + y + 2z = 2 3x + y + 7z = 9