5. Suppose a particle is moving along a smooth path r(t) with constant speed. If a(t) + 0, show that a(to) I (to) (ie. prove that the acceleration vector is perpendicular to the velocity vector).

Q: If a particle moves in the xy-plane so that at time t > 0 its position vector is (In (t2 + 2t) ,…

A: As we know that the expression for the velocity is; v=dstdt Given position vector is:…

Q: 2. Solve the initial value problem for r as a vector function of t. (Hint: think anti-derivative) dr…

A: We will integrate the given function w.r.t. t.

Q: Q2. A particle is moving along the curve whose position vector is given as R(1) = (1) i + ( In…

A: Given

Q: 4. For any time t 2 0, if the position of a particle in the xy-plane is given by x = r² +1 and y=…

A:

Q: 6. A particle moves in the xy-plane in such a way that its velocity vector is (1+1, 1). If the…

A: Find the position vector

Q: Match the vector field with its graph. F(x, y) = yi 址 四 場 x Q 6 電

A: F(x,y)=yi So at any point vector magnitude is same as y-cooridnate Also, if y is negative [below the…

Q: If r(t) = cos(2t)i + sin(2t)j - 4tk, compute the tangential and normal components of the…

A: We have to find tangential component and normal component

Q: 2. Find the r-component of the acceleration for the velocity field of a flow given by v(r, y, z,t) =…

A: NOTE: Hi! Thank you for the question. As per the honor code, we'll answer any one question in case…

Q: 9. An object moving along a curve in the xy-plane has position (x(t),y(t)) at dy dx :=1+ tan(r²) and…

A: According to Bartleby q&a guidelines I will attempt only one question

Q: Suppose that, at time t = 0, a particle with mass 3 has position vector 7(0) = -3 j+ k and velocity…

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: (b) Suppose C is a curve of length N, and ||F|| < M, where M is some positive number. Prove that ·…

A:

Q: Suppose z is given implicitly by the equation ln (yz2) + x3z = 1 in a neighborhood of the point P…

A:

Q: 1. Find one normal vector of the tangent plane to f(x, y) = cos(x² + y²) at point P(1, 1, cos 2).

A: Please check step 2 for the solution.

Q: Suppose that a particle following the path c(t) = (t,t – 5t,0) flies off on a tangent at to = 3.…

A: Given:c(t) = (t2 , t3-5t , 0)

Q: 1. The position of a particle in space at time t is determined by et VI+t: 1 3 j+ t+1 t t a) lim…

A: Solve the following

Q: Match the vector field with its graph. F(x, y) = yi 1 |||||||||| 三 6 靈

A:

Q: give the position vectors of particles moving alongvarious curves in the xy-plane. In each case,…

A:

Q: 5. Is the following vector field F(r, y) = (2x – y, -x + 3y?) conservative? If so, find a potential…

A:

Q: 10. () The gravitational field F(x) = mMG Ex is a conservative vector field.

A: We shall show that F is conservative vector field by using vector calculus please follow next step…

Q: 4. Find the flow of the velocity field F = V(xy²z³) along the line segment from (1,1,1) to (2,1,…

A:

Q: A bee with velocity vector r' (t) starts out at (-4,0,5) at t = 0 and flies around for 5 seconds.…

A:

Q: 2. Show that the given vector functions are linearly dependent on (-0,00). e' x(t) = x, (t) = 4e 2e2…

A:

Q: (a) Is the vector field | F(x,y) = (xe, ye²) conservative? Justify your answer. If it is…

A: Let the vector field F = Pi + Qj.The above vector field is conservative if and only if dPdy=dQdx

Q: A point, p, is moving with acceleration 6t j -k at time t. If at t 1, the coordinates of P are (1,…

A:

Q: Show that the vector field F = x²i + (2yz)j+ y² k is conservative and then find the corresponding…

A: A vector field F is conservative if curl(F)=0 or ∇×F=0 The potential function is given by f such…

Q: Sketch the direction field of the D.E. y = y - t and the solution curves corresponding to y(0) =-1,…

A: Solution of question is written in below steps.

Q: If r(t) = cos(2t)i + sin(2t)j – 4tk, compute the tangential and normal components of the…

A: Given:- rt=cos2ti+sin2tj-4tk Find Tangential and normal component of acceleration vector

Q: 25. A particle is located at (1, 1,0) at time t = 0 and it follows a path whose velocity vector is…

A: Let's find.

Q: 1. The position of a particle in space at time t is de 3 -k: =' - 1 1+t- j+ t+1 a) lim Okelocity…

A: The position vector of the particle is r(t)=et-1ti+1+t-1tj+3t+1k

Q: Find the work done by the force field F in moving an object from P(-3, 7) to Q(7,5). 2x F(x, y) = -

A:

Q: 3. A particle moves in the xy-plane so that at any time t, its coordinates are given by x = t° -1…

A:

Q: the acceleration vector is a(t) = (-1cos(1t))i + (-1sin(1t))j + (1t)k, the initial velocity is v(0)…

A:

Q: 3. Find the length of the trajectory of the vector valued function f(t) = (7V2.1,7",e-7) from 0 to…

A: The detailed solution of the problem is given in the next steps.

Q: (2) Find the line integral of the vector field F(x, y) = (-1,3) along the path from (2, 3) to (2,…

A: The vector field F→=-1, 3To find line integral of vector field along the path 2, 3 to 2, 5

Q: 3. A particle moving along a curve in the xy-plane has position (x(t1),y(t)), with x(1)= 2t +3sin t…

A: Velocity is equal to the first derivative of position vector   Let x =f1t and y =f2t then velocity…

Q: A bee with velocity vector r' (t) starts out at (2, 5, -2) at t = 0 and flies around for 1 seconds.…

A: We can answer first question as per the honor policy. Please resubmit other parts as well to know…

Q: V3 t², a cost + at sin t) 2 r(t) = (a sint – at cos t,

A:

Q: 16. Suppose that over a region of space the electrical potential V is given by V(x, y, z) = 5x² –…

A: We have to find direction in which V change the most rapidly at P and maximum rate of change at P.…

Q: 8. Find the curvature of the vector-valued function r(t) = at the point t = -1.

A: given function is rt=3t2,-4t,1t find the curvature at t=-1 k=r→'t×r→''tr→'t3

Q: 4y The divergence of the vector G = i+ sin y j+ 3k is :- -8y + cosy x3 الخيار 2 O

A:

Q: A ball is bounced directly south, with an initial velocity of 10 m/s off the ground, with an angle…

A: Given initial velocity of ball is 10 m/sec. Angle of elevation is 30°. The ball moves at a constant…

Q: In a horse race, horse 2 passes horse 1 at point A, where the two velocities are v₂ = m 7™ and v₁ =…

A: To determine the distance from point A to B and the position of horse1 relative to horse 2 when…

Q: 6. Compute ſ, F-·dr where f(x, y) = (y² cos x, 2y sin r) and C is the straight line path starting at…

A:

Q: 3. Find the length of the trajectory of the vector valued function f(t) = (7V2 t, e",e-) from 0 to…

A:

Q: Find the derivative of the vector function r(t) = In(4 – 2) i+ v6+ij+7e¬6i k -

A: The given vector function is  r(t)=ln(4-t2)i+6+tj+7e-6tk The objective is to calculate the…

Q: Solve the initial value problem for ř as a vector function of t 1 = 8i + 8j -32k r(0) = 100k dt

A:

Q: 6. Suppose that the temperature T at point (x, y,z) is given by the function T = 2x2 – y? + 4z², and…

A: Given temperature function is: T=2x2-y2+4z2 The tiny bug is at point 1,-2,1 Now, to find in which…

Q: Express the vector field D = ( x2 + y2 )-1 ( xax + yay ) in cylindrical components and cylindrical…

A: