Math 3c Solutions: Exam 1 Fall Graph by eliiminating the parameter; be sure to write the equation you get when you eliminate the parameter.

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1 Math c Solutions: Exam 1 Fall Graph by eliiminating the parameter; be sure to write the equation you get when you eliminate the parameter. x tan t x tan t y sec t y sec t t π 4 To eliminate the parameter, use the Pythagorean identity: 1 + tan t sec t 1 + x y + x y t π 4 } tan {{} t x Eliminating the parameter, we see that we have the following line segment: y + x x Don t forget to include the orientation! When t, we are at the point(,) and when t π 4 we are at the point (,6).. Find the arc length of the polar curve given below. r e θ ; θ ln Page 1 of 1

2 Math c Solutions: Exam 1 Fall 16 r(θ) e θ r (theta) e θ arc length }{{} e θ > so we can drop the absolute value r + (r ) dθ (e θ ) + (e θ ) dθ e 4θ + 4e 4θ dθ e 4θ dθ e 4θ dθ e θ dθ e θ dθ }{{} multiply by 1 in the form }{{} e θ ( dθ) e u du u θ du dθ (eu ) ln (e ln e ) ( 1) () Page of 1

3 Math c Solutions: Exam 1 Fall 16. Suppose u is a unit vector that makes an obtuse angle θ with another vector v as shown in the picture below. v θ π θ p e Express the magnitude of proj u v in terms of the vectors u and v. Do not simply state the correct formula, but use the picture and an appropriate geometric argument to derive it. In the picture above, I ve drawn in the line determined by e, since any projection onto a vector is actually a projection onto the line determined by it. I ve also drawn in the projection, p, in red. Below, I ve drawn the right triangle determined by p and v. v π θ p cos(π θ) cos θ }{{} cosine of theta s supplement is the opposite of cosine of theta p v p v cos θ ( v ) p Page of 1

4 Math c Solutions: Exam 1 Fall 16 cos θ ( v ) ( e ) }{{} 1 }{{} multiplying by 1 does not change the value on the left side of the equation p (v e) }{{} p the dot product of two vectors is the product of their magnitudes and the cosine of the angle between them 4. Let v be the vector in R of length that makes an angle of π 6 with the positive x-axis and let w be the vector of length that makes an angle of π 6 with the positive x-axis. Find the vector component of v along w (that is, the vector component of v that is parallel to w). First, express v and w in component form. v cos π 6, sin π 6, 1, w cos π 4, sin π 4, 1, The vector component of v along w is p proj w v. Page 4 of 1

5 Math c Solutions: Exam 1 Fall 16 p ( v w ) w w w,,,, ( ) , ( ) 1, ( ) 1, 4, 4,. Determine whether the two lines below are parallel, intersecting, or skew: Line 1: x t y 4 + t z Line : r(t),, 1 + t 1,, 4 First, check to see whether the lines are parallel. direction vector for line 1: v 1 1,, direction vector for line : v 1,, 4. The direction vectors are not parallel since they are not scalar multiples of each other; thus, the lines are not parallel. Since the lines aren t parallel, they either intersect or are skew. If they intersect at a point then there must be a value t 1 at which line 1 is at this common point and a value t at which line is at this common point. We set up a system of equations to solve for this common point; if we find a solution, the lines intersect and if we show that there is no solution, the lines are skew. x values same : t 1 + t (1) y values same : 4 + t 1 () z values same : 1 + 4t () Solving equation () for t 1 : 4 + t 1 t 1 4 t 1 Solving equation () for t : 1 + 4t 4 4t 1 t Page of 1

6 Math c Solutions: Exam 1 Fall 16 Checking these values in equation (1): Left side of (1): ( ) Right side of (1): , so there is no solution to our system of equations; the lines are skew. 6. Note that the following questions are grouped together because answering them requires no calculation. They are not necessarily related in topic. For parts (a) and (b), mark each statement True or False. If the statement is false, provide a brief explanation of why it is false. For parts (c) -(e) fill in the blanks. (a) Every plane has exactly two normal vectors. False. While it is true that every plane has exactly directions that are normal to it (for example, with the xy-plane: straight up and straight down) any non-zero multiple of a vector that is normal to a plane is also normal to the plane. Thus, every plane has infinitely many normal vectors. (b), the zero vector in R, is orthogonal to every vector in R. True. (c) If two vectors in R are parallel, then their cross product is. (Be sure your answer is the zero vector, not the zero number!) (d) The work done by a force F acting to move an object in a straight-line path from a point P to another point Q is F PQ. (e) For the polar curve r sin θ, the corresponding parametric equations are: x r sin cos θ sin θ and y r sin θ sin θ sin θ sin θ 7. The two planes given below are parallel. Plane 1: x y + z 4 Plane : x + 4y 6z Find the distance between them. Be sure to show your work; plugging into a memorized formula does not count as showing work. Q( 1,, ) plane p proj npq plane 1 P (4,,) We need a point P on plane 1; I chose to plug in for y and z and then solve for x to get (4,,). We also need a point Q on plane ; again, I chose to plug in for y and z. Solving for x I got ( 1,, ). PQ,, Page 6 of 1

7 Math c Solutions: Exam 1 Fall 16 We can find a vector normal to both planes by reading off the coefficients in front of the variables in either plane. Using plane 1, n 1,, (Note that if we read off the coefficents in front of the variables in plane we get, 4, 6 which is just times n.) The distance between the planes is projn PQ : ( ) PQ proj n npq n n n ( ),, 1,, 1,, 1,, 1,, proj n PQ ( ) 1,, ( ) + ( ) Thus the distance between the planes is Find the equation of the plane indicated in the figure. (As always, only a finite section of the plane is drawn.) Note that I ve written in the coordinates of points on the plane above. We can find by inspection the coordinates of any point that is a corner of the box, so the other point we could find by inspection is (,, ) Point: We can use any of the points; I ve chosen to use P(,,). Normal: Both PQ and PR are parallel to the plane; thus we can calculate a normal vector by taking their cross product. Page 7 of 1

8 Math c Solutions: Exam 1 Fall 16 Point-normal form: PQ,,,, PR,,,, i j k PQ PQ i j + k 4, 6, Let S(x, y, z) represent an arbitrary point on the plane. Then PS x, y, z is parallel to the plane. As such, it is orthogonal to the plane s normal. PS n PS n x, y, z 4, 6, 9. The equation below describes a quadric surface. 4(x ) 6y x 6x + y 4 z 9 We complete the square to put the equation in standard form: x 6x +9 + y 4 z 9 +9 (x ) + y 4 z From this we see that our surface is centered at (,, ). (a) Draw the trace in each of the planes that passes through the center and is parallel to one of the coordinate planes. If you get no trace or just a single point, draw the trace in an appropriate parallel plane/s instead. Note: If you don t use the planes through the center, you can miss the very important information that two of our traces are degenerate hyperbolas that is, intersecting lines! Page 8 of 1

9 Math c Solutions: Exam 1 Fall 16 x (parallel to the yz-plane) y (xz-plane) y 4 z y 4 z y z ± y z (x ) z x z ±(x ) z ±(x ) z ± y z z z y x z (xy-plane) (x ) + y 4 The solution set is the single point (,). y z ± (parallel to xy-plane) (x ) + y 4 1 (x ) + y 4 1 y x x Page 9 of 1

10 Math c Solutions: Exam 1 Fall 16 (b) Sketch the surface in R. (c) What is the name of the surface? elliptic cone 1. (a) Let P be the point (x, y, z) (1,, ). Find the cylindrical coordinates for this point. We are converting from (x,y,z) to (r, θ, z); the z-value will be the same. To find r and θ we can plot the point (1, ) in the xy-plane; we just have to make sure we choose r and θ < π. θ α r Page 1 of 1

11 Math c Solutions: Exam 1 Fall 16 r (1 + ( ) 1 + Letting α reference angle, we see that θ π α. tan α tan θ 1 α arctan( ) θ π α π arctan( ) ( ) Thus (r, θ, z), π arctan( ), (b) Find spherical coordinates for the point in part (a). ρ distance from point to origin x + y + z 1 + ( ) + ( ) θ π arctan( ) since θ is the same is spherical and cylindrical coordinates. r z ρ φ Using the triangle above (and not assuming that φ is acute): Page 11 of 1

12 Math c Solutions: Exam 1 Fall 16 Thus (ρ, θ, φ) z cos φ ρ 6 ( ) φ arccos }{{} reference angle is π 6, cosine is negative, and φ [, π] π 6 (, π arctan( ), π 6 (c) Sketch the constant surface φ π. ) Page 1 of 1

13 Math c Solutions: Exam 1 Fall 16 (d) Sketcht the constant surface θ π. Page 1 of 1