Let u(t)= ti+ln(t)j + et k and v(t) = ti+2tj+1k. Compute the derivative of the dot product [u(t)- v(t)] in two ways and confirm they agree: • Compute the dot product u(t). v(t) first and then differentiate the result. • Alternatively, use the following “Dot Product Rule" v(t)] = u'(t) . v(t) + u(t) . v'(t). (1)
all parts -2t e - (13 points) Let f(t) cos 2t, sin 2t) for t 2 0. F() (a) (4 points) Find the unit tangent vector for the curve d (F(t)-v(t)) using the product rule for dt (b) (5 points) Let v(t) = 7'(t). Calculate the dot product and simplify v(t) (c) (4 points) For an arbitrary vector-valued function 7 (t) with velocity vector = 1, what can be said about the relationship between F(t) and v(t)? if F(t) (t)...
Whats the answer to number 1? 1. Let r(t) = -i-e2t j + (t? + 2t)k be the position of a particle moving in space. a. Find the particle's velocity, speed and direction at t = 0. Write the velocity as a product of speed and direction at this time. b. Find the parametric equation of the line tangent to the path of the particle at t = 0. 2. Find the integrals: a. S (tezi - 3sin(2t)j + ick)...
what is the answer for number 4 1. Let r(t) = -i-e2t j + (t? + 2t)k be the position of a particle moving in space. a. Find the particle's velocity, speed and direction at t = 0. Write the velocity as a product of speed and direction at this time. b. Find the parametric equation of the line tangent to the path of the particle at t = 0. 2. Find the integrals: a. S (tezi - 3sin(2t)j +...
12. Evaluate the integral.∫(sec ²(t) i+t(t²+1)⁵ j+t⁵ ln (t) k) d t13. Find r(t) if r'(t)=t⁵ i+e^{t} j+3te³t k and r(0)=i+j+k.14.Find f'(2), where f(t)=u(t) · v(t), u(2)=(1,2,-1), u'(2)=(3,1,7), and v(t)=(t, t2), t3).
(1 point) Given the acceleration vector a(t) = (-4 cos (2t))i + (-4 sin (2t))j + (-3t) k , an initial velocity of v (0) =i+ k, and an initial position of r (0)=i+j+ k, compute: A. The velocity vector v (t) = j+ . B. The position vector r(t) = j+ k
25 and 27 please 24. u i, v i+j, w i+j+k 36. L 25-26 Use a scalar triple product to find the volume of the parallelepiped that has u, v, and w as adjacent edges. 37. W u = (2,-6, 2), v 〈0, 4,-2), w = (2, 2,-4) to 38. S the vectors lie in the same plane. u=51-2j + k, v=4i-j + k, w=i-j ide 28. Suppose that u (v X w)3. Find (a) u" (w × v) (c)...
(I point) Let F=21+(z + y) j + (z _ y + z) k. (1+4t). y = 4 + 2t, z = _ (1+t). Let the line l be x =- (a) Find a point P-(zo, 30, zo) where F is parallel to 1. Find a point Q (which F and I are perpendicular. Q= and l are perpendicular Give an equation for the set of all points at which F and l are perpendicular. equation: (I point) Let F=21+(z...
Prob 2. Let T be a normal operator on a complex finite-dimensional inner product space V whose distinct eigenvalues are λι, 'Ak E C. For any u E V such that llul-1, show that j-1 for some nonnegative numbers a,, j-1,.,k, that sum up to 1 Prob 2. Let T be a normal operator on a complex finite-dimensional inner product space V whose distinct eigenvalues are λι, 'Ak E C. For any u E V such that llul-1, show that...
let two vectors be a(t) = e^t i + (sin 2t) j + t^3 k and b(t) = (e^-t , cos 3t, - 2 t^3) in euclidean three space R^3. Find d/dt [a(t) * b(t)].