The path r(t) = (t- sin t) i+ (1- cos t) j describes motion on the cycloid x=t- sin t, y = 1- cos t Find the particle's velocity and acceleration vectors at t= and sketch them as vectors on the curve.

Q: Find the work done by the force F = 4xyi + (2x² – 3xy)j by pushing a particle along the line segment…

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Q: You need to find the directional derivative of the functian f(x,y) = tan" (x4) at the point (2,2) in…

A: The given function is fx,y=tan-1xy. The main objective is to find the directional derivative of the…

Q: Find the directional derivative of the function f(, y) = In(x + y°) at the point (- 1,3) in th…

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Q: Find the parametric equations of the tangent line to the curve represented by the vector function…

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Q: Find an equation of the plane tangent to the graph of g(r, y) = 3ev²-z² at the point where (x, y) =…

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Q: Find a vector which is tangent to the curve of intersection of the surfaces z = x² – y? and xyz + 30…

A: We will find out the required tangent vector.

Q: Calculate the velocity and acceleration vectors and the speed at the time indicated.r(θ) =〈sin θ,…

A: Consider that: rθ=sin θ, cos θ, cos 3θ at θ=π3. Velocity vector denoted by vθ is vθ=r'θ.  Calculate…

Q: At time to, a moving particle has the velocity vector v = 7i and the acceleration vector a = 7i +…

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Q: 0. for the funchion f (x,4) = 3-x?- u?, find a unit tangent vector to the level curve at the point…

A: Given: fx,y=3-x2-y2-3 at P3,5​​​​​​​

Q: Sketch the curve parametrized by r(f) = (1 – 12, 1) for -1sı<1. Compute the tangent vector at i = 1…

A: Let's first draw the curve -  Domain - -1 ≤ t ≤ 1 where, we can take values of t as -  t = -1, -0.5,…

Q: Find an equation for the tangent line to the curve given parametrically by x(t) 2t, y(t) 4t2 +t+ 3 =…

A: we want to find the tangent lines to theparametric equations given byx=f(t)  y=g(t)The equation of…

Q: Compute the directional derivative of g(x,y)=sin(pi(5x-5y)) at point P(1,-3) in the direction…

A: Firstly, we calculate unit vector in direction of vector u.

Q: Find equations of the Normal plane and Osculating plane of the curve x = 2 sin 3t , y = 2t, z =…

A: Given that x=2 sin3t , y=2t and z=-2cos3t and point 0,π,-2 find equations of the normal and…

Q: ) Find the orthogonal trajectories of the family of curves r = 3C sin 0, where C being parameter.

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Q: 8. Determine the equation of the plane tangent to z point (x, у, z) 3 (-2, 1, 4). x*y3 at the =

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Q: Find the directional derivative of the function f(x,y, z) = V22+ y² + z² at point (2, -3, 6) in the…

A: In this question, we want to find the directional derivative of the given function at given point…

Q: Find the equation of the tangent plane to the surface 3 – 12y – sin(r*:) = 0 at the point (2, 1,0).…

A: To find gradient of the function.

Q: Find the direction in which the maximum rate of change occurs for the function f(x, y) = 3x sin(xy)…

A: To find the Direction in which the maximum rate of change occurs for f(x,y)=3xsin(xy) at point (1,3)…

Q: Find parametric equations for the tangent line to the curve of intersection of the cylinders x = 74…

A: given curves x2+z2=74 , y2+z2=74 , point 5,-5,7  to find parametric equations for the tangent line…

Q: Find the unit tangent vector of the given curve. r- (0)%= (4 cos(금)) i-+ (4 sin()) 」--121 k-. %3D…

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Q: Find a vector parametrization of the circle of radius 3 in the xy-plane, centered at (4, -1),…

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Q: Find an equation of the plane tangent to the surface sin(xyz) = at the point (T, 1,-)

A: sinxyz=12 at the point π, 1, 16. Consider, sinxyz=12⇒xyz=sin-112⇒xyz=π6⇒z=π6xy Differentiating…

Q: Find the tangent line(s) to the parametric curve given by a = t-9t and y =t at (0, 9) %3D %3D

A: We have to find the tangent line

Q: 4. Determine the directional derivatives of $(x, y) = tan"2+3 sin ). at the point (1,1) in the…

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Q: an object is moving in 3-dimensional space with velocity vector v(t)=(sin(t),t,t^2) starting at…

A: Consider the given velocity vector: vt=sint,t,t2 Find the position vector: Integrate velocity vector…

Q: Find the vector equation of the tangent line to the curve given by 7(t) = sin ti + cos (2t)j + tan…

A: There is no such introduction for the solution.I have explained it within the solution.Please go…

Q: Suppose that a motion described by the vector valued function (i.e. position function) r(t) has…

A: Given that :- r't=vt=5cost , 5sint ,-2 and r0=2,1,,1 find position function rt

Q: Find the direction in which the maximum rate of change occurs for the function f(z,9)2zsin(ay) at…

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Q: Find the direction in which the maximum rate of change occurs for the functio f(r, y) = 2x sin(ry)…

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Q: Find the equation of the osculating plane of the helix x = sin 2t, y = t, z = cos 2t at the point…

A: The helix is given as x=sin 2t, y=t, z=cos 2t. We have to find the equation of the osculating plane…

Q: Find the maximum rate of change of f(x, y) Maximum rate of change: = In(x² + y?) at the point (3, 1)…

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Q: Find the two-unit vectors that are parallel to the tangent line to the curve y =x2sin(x) at the…

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Q: A particle moving along a curve in the xy-plane has position (x(1),y(t)) at time dy dx t with dt =…

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Q: Compute the directional derivative of f (x,y)= at the point (1,1) in a direction normal to the…

A: The directinal derivative is given below

Q: Find the parametric equations of the tangent line to the curve of intersection of the surfaces x² -…

A: The equation of the tangent plane to the surface f(x, y, z)=0 at x0, y0, z0 is defined as: fxx0, y0,…

Q: The path r(t) = (t– sin t) i + (1- cos t) j describes motion on the cycloid x=t- sin t, y = 1- cos…

A: Given:           r(t)=(1−sint)i+(1−cost)j

Q: Find the tangent plane to the equation z 2y cos(5x – 4y) at the point (4,5, 10)

A: Given; z=2ycos(5x-4y) at the point (4,5,10)

Q: Find the points (x, y) on the curve with the parametric equations (x3Dt3- 4t at ly=t +2 which the…

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Q: Find the parametric equations of normal line to the surface given by f(x, y) = tan v x2 +y2 at the…

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Q: The directional derivative of xy f(x, y) = x + y) at (1, 1) in the direction of the unit vector at…

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Q: Choose the direction vector for which the function f defined by f(x, y)= sin(3x) cos(11y) has the…

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Q: 16. Sketch the cycloid r(t) = (t- sint, 1 - cos t) together with its tangent vectors at t = and .…

A: Solved the above problem.

Q: Let f(x, y) = sin(cy). Then the directional derivative of f in the direction of the vector at the…

A: fx,y=sinxy We have to confirm whether directional derivative of f in the direction of the vector…

Q: We have to find parametric equation for a line that is tangent (intersection) of the two fields…

A: We have to find parametric equation for a line that is tangent (intersection) of the two fields…

Q: The path r(t) = (t- sin t) i+ (1- cos t) j describes motion on the cycloid x=t- sin t, y=1- cos t.…

A:

Q: Find the work done by the force F = 4xyỉ + (2x2 - 3xy)j by pushing a particle along the line segment…

A: Step:-1 Given Force is F→ =4xy i→ +2x2 -3xy j→ Here, given that particle moves along the line…

Q: Find the orthogonal trajectories of the family of curves r 5C sin 0, where C being parameter.

A: Follow the below steps to find the orthogonal trajectories. Differentiate the function with respect…

Q: Find the divergence and curl of the vector A = rz sin ø ↑ + 3rz² cos o and evaluate both answers at…

A: Consider the given vector. A=rzsinϕr^+3rz2cosϕϕ^ The divergence formula is defined as. divF=Px+Qy…

Q: nit vector that is parallel to the

A: Here have to find the unit vector that is parallel to the line tangent to the parabola y = x2 at the…

Q: Find the velocity vector for the position vector r(t) = (sin(3t), 10t°, e -"). x component = y…

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Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.