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- 1. A body rotates with a constant angular velocity w = w,î + wzĵ + w3k about an axis. Let ř be the position vector of a point P on the body measured from the origin. Then the linear velocity i of the rotation is i = w x i if ř = xî + yî + zk. Show that w =- curl v . 28) Find the position vector r(t) for a particle with acceleration a(t) = (5t, 5 sin t, cos 6t), initial velocity (0) = (3, -3, 1) and initial position (0) = (5, 0, -2).Let u=2i-j, v = 5i+j, w = i +5j Find the specified scalar. (4u) v (4u) v =
- 6. Find the parametric equations of the line tangent to the curve C defined by the vector equation j() = (V2 sin t)t – (cos 2t)J + 4tk at the point where t =4. What is the symmetric form of this line?Sketch the curve whose vector equation is Solution r(t) = 6 cos(t) i + 6 sin(t) j + 3tk. The parametric equations for this curve are X = I y = 6 sin(t), z = Since x² + y² = + 36. sin²(t) = The point (x, y, z) lies directly above the point (x, y, 0), which moves counterclockwise around the circle x² + y2 = in the xy-plane. (The projection of the curve onto the xy-plane has vector equation r(t) = (6 cos(t), 6 sin(t), 0). See this example.) Since z = 3t, the curve spirals upward around the cylinder as t increases. The curve, shown in the figure below, is called a helix. ZA (6, 0, 0) (0, 6, 37) I the curve must lie on the circular cylinder x² + y² =Let u = 5i - j, v = 3i +j, w=i+2j Find the specified scalar. (4u) • v (4u) • v =
- Consider the wave equationutt = uxx, (x, t) ∈ R2. Find two disctinct solutions to the equation.Find the velocity vector in terms of u, and ug. T=2 cos 2t and 0= 4t WO A. V= (4 sin 2t)u, + (8 cos 21)u, O B. v= (-4 sin 2t)u, + (8 cos 2t)u, OC. v-(-2 sin 2t)u, + (8 cos 21)ug O D. v= (-8 sin 4t)u, + (4 cos 4t)u, Click to select your answer. 2Type here to search3. Let vector à = a, +2r cos Øä, +3ä. (a) (b) (c) (d) Express the vector à in rectangular coordinates. Express the vector A in cylindrical coordinates. Find A (2,0,1). ind Ã(4,73,-1).
- Consider the wave equation utt = uxx, (x, t) ∈ R2. find 2 distinct solutions please4. Find the vector equation for a particle if A(t) = 6ti + 12t²j+ k, V(0) = 2i + 3j, and R(0) = 4k.Exercises 9–12 give the position vectors of particles moving along various curves in the xy-plane. In each case, find the particle's veloc- ity and acceleration vectors at the stated times and sketch them as vec- tors on the curve. 9. Motion on the circle x² + y? r(t) = (sin t)i + (cos t)j; t = T/4 and /2 10. Motion on the circle x? + y² = 16 r) = (4 cos )i + (4 sin i: r(t) = ( 4 cos t = T and 37/2 11. Motion on the cycloid x = t – sin t, y = 1 – cos t r(t) = (t – sin t)i + (1 – cos t)j; 12. Motion on the parabola y = x² + 1 r(t) = ti + (t² + 1)j; t= -1,0, and 1