Prove that, for the system
Hint: Let
Want to see the full answer?
Check out a sample textbook solutionChapter 7 Solutions
Differential Equations: An Introduction to Modern Methods and Applications
Additional Math Textbook Solutions
Calculus: Early Transcendentals (2nd Edition)
Elementary Statistics Using The Ti-83/84 Plus Calculator, Books A La Carte Edition (5th Edition)
College Algebra with Modeling & Visualization (5th Edition)
A First Course in Probability (10th Edition)
Algebra and Trigonometry (6th Edition)
Basic Business Statistics, Student Value Edition
- Show that the three points (x1,y1)(x2,y2) and (x3,y3) in the a plane are collinear if and only if the matrix [x1y11x2y21x3y31] has rank less than 3.arrow_forwardConsider an arbitrary program in standard equation form: maximise cTx subject to Ax = b, x 2 0 Show that if 1) and z are two different optimal solutions to this program, then every convex combination of y and z is also an optimal solution of the program. 2) Consider two vectors y and z with y z and consider two values 0 E [0, 1 and 0 e [0, 1] with 0 0. Show that Oy + (1 – 0)z 0'y + (1 – 0')z. In other words, every distinct way of choosing the value A E [0, 1] gives a distinct convex combination Ay + (1 – A)z. Hint: let a = Oy + (1-0)z and b 0'y + (1- 0)z, then consider a - 3) Using the result from parts (a) and (b), show that every linear program has either 0, 1, or infinitely many optimal solutions.arrow_forward4) Come up with two vector functions r₁(t) and r₂(t) that are non-linear and satisfy the following: The functions have an intersection point at (2, 2, 0), but it is not a collision point. At the intersection point, the function's paths are orthogonal.arrow_forward
- Suppose you are a bug buzzing around in space. You are currently at the point (1, 1, 0). If the temperature at point (x,y,z) is given by T(x, y, z) = xyz + x²y Z. If you are a bug that likes warm temperatures, which direction should you fly in to move in the direction of most rapid temperature increase? (0, 1, 2) (0, 2, 1) (1, 0, 2) ○ (1,2,0) (2,0,1) (2, 1,0)arrow_forward2. Consider the linear system x'(t) 1 2 x(t) defined for t > 0. Verify that the following - (; +2) 1+] pair of vector-valued functions form a fundamental set of solutions. tet x1(t) = | X2(t) =arrow_forwardShow that the line x + 1 =y/−1=(z − 2)/2 is in the plane 2x + 4y + z = 0.arrow_forward
- please answer allarrow_forwarda) Compute the work done by F = (y², sin(z), x) along a straight line from (0,3,0) to (1, 0, 1). b) Compute the work done by F = (y², sin(z), x) along a triangular path from (0, 3, 0) to (1, 0, 1) to (0, 0, 2) and back to (0, 3, 0). c) Compute the work done by = (x, y², sin(z)) along a circular path of radius 3, centered at (0, 1, 0), in the plane y = 1, counterclockwise when viewed from the origin.arrow_forwardf(x. y.z) = + xy + yz + Q8) Find the Linearization at (1,1.2)arrow_forward
- Consider 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).arrow_forwardGiven A = (a1, a2, az) and B = (b1, b2, b3) a parametrization of the segment line joining B to A is given by (a)X(t) = ((b, – a,)t + a4, (bz – az)t + az, (bz – az)t + az) for t E [0, 1] (b)X(t) = ((a, – b,)t + b,, (az – bz)t + b2, (az – b3)t + b3) for t E [0,1] (c)X(t) = ((b, – a,)t + b,, (b, – az)t + b2, (bz – a3)t + b3) for t € [0, 1] (a) O (b) O (c)arrow_forwardWritten Problem 3: An Acura and a BMW are driving around an empty planar parking lot, represented by the xy-plane. The trajectories of the cars are given parametrically by A(t) =(5t +3, – 21) (the Acura), B(t)=| 21 +- 15 ,t² + 2 2 33 -* (the BMW), 4 where t represents time in minutes. a) Show that these cars crash into one another at a certain time. b) Compute the slopes of the tangent lines to the cars' trajectories at the time of the crash. Give your answers in fraction form. c) If you did part (b) correctly, you should notice something geometrically interesting about the tangent lines. What do you notice, and what does it tell you about how the cars crashed (ex: did they crash head-on)?arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning