Module 3 Lab Assignment Lines, planes, and surfaces in space Instructions: These problems are meant to be completed in collaboration with your classmates. You may choose to collaborate face-to-face or online. See Canvas for more details. Some problems are more challenging than those covered in class. These are called Challenge Problems and they are related to class material but in a manner you may not have seen during class. Don't worry; any reasonable attempt on a Challenge Problem will receive full credit. The intention is to challenge your problem-solving abilities! Challenge Problems are marked with a (There is no challenge problem on this lab.) a Consider the point P(-1,3,1). Use this point to answer questions 1 through 6 on this assignment. 1. Find an equation for a line containing P and passing through the origin. P = ( - 1 . 3 . 1 ) origin = ( 0.0, 0 ) "(6 ) = = 17 + ) = 1 0 . 0 , 0 ) + 8 ( - 1 , 3 , 1 ) 2 . Find an equation for a line which contains P and the point Q ( 2, - 1, 3 ) rattle Rvector form = ( Q - P > "( t ) = ( - 1, 3, 1 ) + t ( 3, - 4, 2 ) = ( 2 - 1, 3 ) - ( - 1.3 1 ) = (2+1 . - 1 - 3 , 3 - 1 ) ( - 2 + 3 t . 3 - 46 , 1 1+ 2t ) b = ( 3 , - 4, 2 ) X = - 1+3t = 3-42 2 = 1+2 t 3. Find the equation for the plane containing the two lines you just found.4. Find an equation for the plane that contains P and is perpendicular to the y-axis. 5. Find the equation for a plane containing P which is perpendicular to the line y = x. 6. Find the equation for a plane which does not contain P 1 + 2 tConsider the surface given by x2 + y2 - z2 = a. Graph this in GeoGebra then answer questions 7 through 10 using your online graph. Note: You will need to change the value for the slider given for a to a minimum of -20 a maximum of 20 with an increment of 1. To do so, follow the instructions found in this video. 7. What quadratic surface do you obtain when a > 0? 8. Describe what occurs as the slider moves from a = 20 to a = 0 9. Explain what occurs when a = 0. Does this make sense? Why or why not? 10. Describe what occurs when a