Graph the following quadratic equations in R and identify them as either a parabola, hyperbola, or...

 

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Graph the following quadratic equations in R and identify them as either a parabola, hyperbola, or an ellipse. All work to find intercepts must be shown; do not use a calculator or computer. (a) x + y = 5 (b) 9x + y = 9 (c) x - y = 4 (d) x + y - 6x - 2y + 1 = 0 (e) x + y +3=0 (f) 4x - y = -16 3. Let v be the vector from (2, 5, 3) to (6, 1,5). (a) Find ev, the unit vector pointing in the same direction as v. (b) Find a vector of length 5 in the direction opposite to v. 4. Consider the three vectors in R: u = (1,1), v = (4,2), w = (1,-3). For each of the following vector calculations: [P] Perform the vector calculation graphically, and draw the resulting vector. (Draw a labeled and to-scale graph.) Calculate the vector calculation arithmetically and confirm that it matches your picture. (a) 3u + 2w (b) u + v + w (c) 2vw7u 5. [P] Draw a graph of the following systems of equations in R. How many dimensions does the graph have? Classify the graph as a point, a curve, or a surface. (a) (a) { +0 x + y 9 (b) = 0 = 4 (b) AV +y +2=4 |z0 x (c) 6. [P] Draw a graph of the following systems of equations in R. How many dimensions does the graph have? Classify the graph as a point, a curve, a surface, or a solid. x = 5 (c) y=-3 (e) 2x + 3y = 6 x - 2y = 4 (f) x + y = 16 z = 2 z = 3x - y (d) x + y + z 1 7. Find the center and radius of the sphere given by the equation x + y + z + 4x 6z+ 4 = 0. What is the highest point on this sphere? x + y 16 = 2 (d) Sy x x (h) 11. A drone starts at (3, 1, 4) and flies in a straight line in the direction of (-2, 3, 1). (a) Does the drone pass through the point (-3, 10, 7)? If so, with what t-value? (b) Does the drone pass through the point (-1,7, 3)? If so, with what t-value? Using tip-to-tail addition We mean "highest" in the normal sense, which would be the largest z-coordinate. x + y = 16 [0 12. Line E goes through the points (1,5,2) and has the direction vector (2,2, 3). Line F goes through the points (3, 1,6) and (5, 0, 2). Are these lines parallel? If not, do these lines intersect? 13. Particle 1 starts at point (3,1,1) and particle 2 starts at point (10,5, 4); at t = 0, both particles begin to move along linear paths. After 1 second, particle 1 is at point (4, 3, 4) and particle 2 is at (8,6, 3). (a) Do the paths of the particles intersect? (b) Will the particles collide? 14. (*) A laser pointer is situated at the point (3, 1, 4) and is pointing a linear beam in the direction (-1,2,-2) towards the plane 4x - y - 5z = 7. On the plane is a target at the point (1, 12, -3). (a) Does the laser hit the target? If not, where does the laser hit the plane, and how far away is this from the target. (b) [C] How far off is the angle of the laser from hitting the target? Give your answer in degrees, rounded to 3 decimal places. 15. Consider a mirror lying along the x-axis in R. A laser starts at the point (2, 6) and is fired in the direction of (4, -3). The beam travels until it hits the mirror, where it reflects off the mirror and then travels until it hits a wall on the line x = 20. (a) Give a parametrization for the initial path of the laser. (b) At what point does this beam hit the mirror? (c) Give a parametrization for the path of the reflected beam. (d) At what point does this beam hit the wall? 16. (*) A billiards table is 8 feet by 4 feet, and has pockets in all four corners and in the middle of the two longer sides; we can visualize this as a rectangle in R and assume that the bottom left pocket is at the origin and the top right pocket is at (8,4). A ball starts at the point (1, 2) and is hit in the direction (5,-2). Does the ball go into a pocket before it bounces off three walls and, if so, which one? 17. Are the following statements true or false? Explain why or why not. (a) A vector has only one unit vector parallel to it. (b) There is only one parametrization for a line or line segment. (c) The sphere x + y + z = 1 is a three-dimensional surface. (d) If u + v=w, then ||u|| + ||v|| = ||w|| Graph the following quadratic equations in R and identify them as either a parabola, hyperbola, or an ellipse. All work to find intercepts must be shown; do not use a calculator or computer. (a) x + y = 5 (b) 9x + y = 9 (c) x - y = 4 (d) x + y - 6x - 2y + 1 = 0 (e) x + y +3=0 (f) 4x - y = -16 3. Let v be the vector from (2, 5, 3) to (6, 1,5). (a) Find ev, the unit vector pointing in the same direction as v. (b) Find a vector of length 5 in the direction opposite to v. 4. Consider the three vectors in R: u = (1,1), v = (4,2), w = (1,-3). For each of the following vector calculations: [P] Perform the vector calculation graphically, and draw the resulting vector. (Draw a labeled and to-scale graph.) Calculate the vector calculation arithmetically and confirm that it matches your picture. (a) 3u + 2w (b) u + v + w (c) 2vw7u 5. [P] Draw a graph of the following systems of equations in R. How many dimensions does the graph have? Classify the graph as a point, a curve, or a surface. (a) (a) { +0 x + y 9 (b) = 0 = 4 (b) AV +y +2=4 |z0 x (c) 6. [P] Draw a graph of the following systems of equations in R. How many dimensions does the graph have? Classify the graph as a point, a curve, a surface, or a solid. x = 5 (c) y=-3 (e) 2x + 3y = 6 x - 2y = 4 (f) x + y = 16 z = 2 z = 3x - y (d) x + y + z 1 7. Find the center and radius of the sphere given by the equation x + y + z + 4x 6z+ 4 = 0. What is the highest point on this sphere? x + y 16 = 2 (d) Sy x x (h) 11. A drone starts at (3, 1, 4) and flies in a straight line in the direction of (-2, 3, 1). (a) Does the drone pass through the point (-3, 10, 7)? If so, with what t-value? (b) Does the drone pass through the point (-1,7, 3)? If so, with what t-value? Using tip-to-tail addition We mean "highest" in the normal sense, which would be the largest z-coordinate. x + y = 16 [0 12. Line E goes through the points (1,5,2) and has the direction vector (2,2, 3). Line F goes through the points (3, 1,6) and (5, 0, 2). Are these lines parallel? If not, do these lines intersect? 13. Particle 1 starts at point (3,1,1) and particle 2 starts at point (10,5, 4); at t = 0, both particles begin to move along linear paths. After 1 second, particle 1 is at point (4, 3, 4) and particle 2 is at (8,6, 3). (a) Do the paths of the particles intersect? (b) Will the particles collide? 14. (*) A laser pointer is situated at the point (3, 1, 4) and is pointing a linear beam in the direction (-1,2,-2) towards the plane 4x - y - 5z = 7. On the plane is a target at the point (1, 12, -3). (a) Does the laser hit the target? If not, where does the laser hit the plane, and how far away is this from the target. (b) [C] How far off is the angle of the laser from hitting the target? Give your answer in degrees, rounded to 3 decimal places. 15. Consider a mirror lying along the x-axis in R. A laser starts at the point (2, 6) and is fired in the direction of (4, -3). The beam travels until it hits the mirror, where it reflects off the mirror and then travels until it hits a wall on the line x = 20. (a) Give a parametrization for the initial path of the laser. (b) At what point does this beam hit the mirror? (c) Give a parametrization for the path of the reflected beam. (d) At what point does this beam hit the wall? 16. (*) A billiards table is 8 feet by 4 feet, and has pockets in all four corners and in the middle of the two longer sides; we can visualize this as a rectangle in R and assume that the bottom left pocket is at the origin and the top right pocket is at (8,4). A ball starts at the point (1, 2) and is hit in the direction (5,-2). Does the ball go into a pocket before it bounces off three walls and, if so, which one? 17. Are the following statements true or false? Explain why or why not. (a) A vector has only one unit vector parallel to it. (b) There is only one parametrization for a line or line segment. (c) The sphere x + y + z = 1 is a three-dimensional surface. (d) If u + v=w, then ||u|| + ||v|| = ||w||