Name TEST 1 answers No Calculators 1 48 points A hummingbird starts at a feeder located at F 0 0 10 distances in feet and flies straight toward a point B 10 20 30 on a branch with a speed 3f t s An observer is located at O 5 10 10 a 5 How long does it take the hummingbird to reach the branch F B 100 400 400 30 F B 10 20 20 30 F B 10 seconds travel time speed 3 b 5 What is the hummingbird s velocity vector 1 v speed unit direction vector 3 F B 1 2 2 F B c 5 Write down the equation of the line that contains the hummingbird s path r r0 vt where r0 0 0 10 x t y 2t is the starting point hence z 10 2t d 5 Where is the hummingbird going to be in 2 seconds x 2 y 2 2 4 z 10 2 2 14 e 7 What is the projection of F O onto F B 25 50 50 F B F O 25 1 2 2 proj F O F B FB 9 9 9 9 F B 2 f 7 What is the equation of the plane that contains the triangle 4F OB i j k F B F O 10 20 20 200 100 0 normal to the plane 5 10 0 0 0 10 is a point in the plane hence 200 x 0 100 y 0 0 z 10 0 or 2x y 0 1 2 3 4 5 g 7 What is the area of the triangle 4F OB 1 1p area F B F O 2002 1002 50 5 2 2 h 7 How close to the observer will the hummingbird get using g F B F O 100 5 distance 30 F B using e distance F O proj F O s FB 25 5 9 2 50 10 9 2 50 9 2 10 5 3 2 24 points a Find the parametric equation of the line through the point P 1 0 1 and perpendicular to the plane 5x 2y 3z 7 normal to the plane v 5 2 3 line x 1 5t y 0 2t z 1 3t b Find the point R where this line intersects the plane finding t such that x 1 5 1 5t 2 2t 3 1 3t 7 25 63 38 38 gives t 10 15 23 z 1 38 38 38 63 10 23 R 38 38 38 y c Find the distance of the point P from the plane using b distance P R s 63 1 38 2 10 38 2 23 1 38 2 5 38 as in the book p 886 choosing a point Q 1 1 0 on the plane gives distance P Q v 5 v 38 5 38 3 18 points Find an equation of the plane that contains the origin and the line y 1 t x 1 2t z 2t t vector parallel to the line v 2 1 2 point on the line t 0 Q 1 1 0 position vector u 1 1 0 normal to the plane i j k v u 2 1 2 2 2 3 1 1 0 plane 2x 2y 3z 0 4 10 points Sketch and identify the surface x2 y 2 z 2 1 0 In the plane z c we have a circle x2 y 2 c2 1 radius c2 1 hence z has to be 1 So we have hyperboloid of two sheets p 895 5 10 extra credit points if you get 90 or more points on problems 1 4 points Find the projection of the line r r0 tv t onto a plane which contains a point with a position vector r1 and has a normal n Hint Start by following the first steps in problem 2 Let P t be a point at a fixed t on the line The equation of the line perpendicular to the plane and passing through P t is r r0 tv sn s It intersects the plane when r0 tv sn r1 n 0 hence s r0 tv r1 n n 2 Hence the projection of P t on the plane is r0 tv r0 tv r1 n n n 2 which represents a line with a direction v v n n v projn v n 2 and passing through a point r0 r0 r1 n n n 2