in 4 method ТВС- тво-1 Example 1.8.6 Find the p sic equations of the plane that passes through the three points (x, y, z) = (-1, 2, 1), (x, y, z) = (1, 2, 3) and (x, y, z) = (2, -1, 5).
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- a)Find the equation of the normal line at the point (−3,2,6) to the ellipsoid x^2/ 4 + y^2+ z^2 /9 = 3. b) Find a plane through P (3,5,1) and perpendicular to the line of intersection of the planes: 2x+y−z = 3 and x+2y+z = 2. c)Classify all critical points for f (x,y) = 2x^2 −y^3 −2xyConsider a point particle with position vector r = (x, y, z) in Cartesian coordinates, moving with a velocity v = (β, αz, −αy), where α and β are positive constants. Find the general form of r(t), the position of the particle, as a function of time t, (hint: write v = (β, αz, −αy) as a system of first order ODEs and note that the equation for x is decoupled from the others). Describe in words the motion of the particle and sketch its trajectory in R3 (you can use software packages for the plot).1. The distance of a point in the 3-D system from the origin a. is defined by the absolute value of the vector from the origin to this point. b. is the square root of the square of the sums of the x-, y- and z-values. c. is the square root of the sum of the squares of x-, y- and z-values. d. can either be negative or positive. e. None of the above. 2. In parametrizing lines connected by two points in 3-D plane, a. there is only one correct parametrization. b. symmetry equations may not exist. c. a, b, and c must not be equal to 0. d. the vector that connects the two points is a scalar multiple of the vector containing the direction numbers. e. None of the above. 3. A plane in 3D-space system a. is generated by at least three points. b. can lie in more than one octant. c. must have a z-dimension. d. must have a point other than the origin. e. None of the above. 4. A quadric surface a. must have either x2, y2, or z2 or a combination of those, on its general expression. b. must have a…
- Jackie and Christine are racing again! We only get some discrete data this time, but we can find the details through calculation! 3 minutes after the race starts, Jackie is 12 miles away from the starting point, and Christine is 27 miles ahead of Jackie. 13 minutes after the race starts, Jackie is 52 miles away from the starting point, but now Christine is only 17 miles ahead from Jackie. Assume they are going at a constant speed. (a) Present their movement on the xy-plane, where the x- and y-axis represent the time since the race starts and their distances from the starting point respectively. Label the four points clearly. (b) Find the speed of Christine and Jackie. You may use the unit “miles per minute” or, if you are comfortable with unit conversion, “miles per hour.” (c) Is anyone given a head start? If so, who is that and how far was that? (d) How far is Jackie from the starting point when he catches up with Christine?Consider the trajectory of a particle on an (x, y) plane. Let (Xn, Yn) be the position of the particle at the nth step, n ≥ 0. The particle is initially at position (0, 0) i.e., X0 = 0 and Y0 = 0. Let its sequential movements, i.e., between steps n and n + 1, be defined as follows: P(Xn+1 = i−1, Yn+1 = j|Xn = i, Yn = j) = P(Xn+1 = i+1, Yn+1 = j|Xn = i, Yn = j) = 1/4, P(Xn+1 = i, Yn+1 = j−1|Xn = i, Yn = j) = P(Xn+1 = i, Yn+1 = j+1|Xn = i, Yn = j) = 1/4. (a) Are Xn and Yn independent? (b) Calculate Cov(Xn, Yn).find a parametrization for the curve. 1. the left half of the parabola y = x2 + 2x 2. the ray (half line) with initial point (2, 3) that passes through the point (-1, -1) 3. the ray (half line) with initial point (-1, 2) that passes through the point (0, 0)
