If a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t), show that the curve lies on a sphere with center the origin.

Q: What is the unit tangent vector to the curve r(r)=sint i+2cost j when t=π/4

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Q: r(t) = (sin 7t, t, cos t)

A: The vector equation is r(t)=sinπt, t, cosπt. Here, x=sinπt, y=t, z=cosπt Then x=sinπy, and z=cosπy…

Q: Find the ordinate of the center of curvature of the curve y² = 4x at point (2, 4) from the x-axis.

A: Given option are wrong, option c should be '-16'.

Q: Find the curvature and radius of curvature of the plane curve at the given value of x.

A: We have,   It is known that radius of curvature is given as, Here K is curvature of the curve.…

Q: -*) at (0, 1). Find the center of curvature of the curve

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Q: A unit tangent vector to the curve r(t) = [sin(t), cos(t), 2f] at the point A:(0, — 1, 2п) is given…

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Q: Find the curvature of the curve defined by F (+) = (5 cos(t), 6 sin(t)) at t = 2

A: κ=  r'(t)→× r"(t)→r'(t)→3 We have to just evaluate for given r(t)

Q: find the equation of the tangent plane of this surface

A: In this question, we check the surface is simple and then we find the equation of tangent plane at…

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

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Q: Find an equation of the plane tangent to the surface

A: Given that : The equation is x2 + y 2 z 2 = 8 and the point P = ( 2, 2, 1).

Q: find the curve’s unit tangent vector. Also, find thelength of the indicated portion of the curve. .…

A: Given:                  rt=cos3tj+sin3tk,     0≤t≤π2                 v=ddtrt=ddtcos3t j+sin3t…

Q: Ř(u, v) = (u², v – 2u, u + v²)

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Q: How would I find the unit tangent vector T(t) at the point with the given value of the parameter t.

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Q: A normal vector to the tangent plane for the at the point (1, 1, 1) is: y surface x =

A: A(1, 1, 1)z= xyf(x,y,z) =xy-zfx=yfy=xfz=-1Normal vector to the tangent plane for the surface;n^…

Q: Find the unit tangent vector of the curve: 7(t) = (3 cos t , 3 sint, t2) at t = 0. %3D

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Q: At an air show, a jet has a trajectory following the curve y = x2. Ifwhen the jet is at the point…

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Q: Find the angle of inclination of the tangent plane to the surface at the given point. x² + y² = 5,…

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Q: Consider the following curve 7(1) = (-2sin(e), –V5,2cos() (a) Find the unit tangent vector. (b) Find…

A: The formula to obtain the unit tangent vector for any curve r(t) is given by: The formula to find…

Q: Find the unit tangent vector to the curve defined by F(t) = (4 cos(t), 4 sin(t), 3 sin (t)) at t = 3…

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Q: Find a unit vector parallel to the tangent line of y = x at (2,8). What angle does this vector make…

A: Given that, y=x3   We have to find a unit vector parallel to the tangent line of the given curve at…

Q: Find parametric equations for the tangent line to the curve a-tcost, y-t, z- tsint at the

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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: Find a vector parametrization of the circle of radius 5 in the xy-plane, centered at the origin,…

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Q: Find the equation of the tangent line of the vector function F(t) = (e= cos(t), e=' sin(t), e') at…

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Q: Find the curvature and radius of curvature of the plane curve at the given value of x. y = sin 2x, x…

A: Consider the provided equation, y=sin2x,   x=π4 Find the curvature and radius of curvature of the…

Q: for the follawing , find the unit tangent vector: parameterized curve rit) =<2 sint) , 2 cos lt),5…

A: Given: r(t)=2sin(t),  2cos(t), 5cos(t),  0≤t≤π The unit tangent vector to curve r(t) is…

Q: Find a vector parametrization of the circle of radius 5 in the xy-plane, centered at the origin,…

A: Solution: Given, Circle having center at (0, 0) and radius is 5 and

Q: Write a formula for the curvature of a plane curve with equation y= f(x)

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Q: Find the curvature and the radius of curvature at any point on the curve y = In sec x.

A: A curve given y = ln (secx)  Find out the curvature and the radius of the curvature of given curve.

Q: Find a unit tangent vector to any point on the curve x = a cos wi, y = a sin wt, z = bt where a,b,w…

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Q: Find the unit tangent vector to the curve defined by =(- 4t, 4t, V49 - t) at t = - 6. T(- 6)

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Q: At which point P of the curve defined vectorially with r (t) = (21, 4t + 1, t) is the tangent vector…

A: The general form of the parametric equation of the curve in three dimensional space is given by…

Q: Find an equation of the tangent(s) to the curve with paramnetric equations ¤ = 1+ VE, y = e" at the…

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Q: Find the radius of curvature of the curve generated by the position vector r (t) = (-40 sin(t),–9t,…

A: Given, r(t)=-40sint,-9t,40cost r'(t)=-40cost,-9,-40sint r'(t)=-40cost2+(-9)2+-40sint2=41…

Q: Find the curvature of the plane curve y = -t at the point t = 2. к(2) -

A: Given:y = -t4differentiating with respect to t, we get:            y' =     d(-t4)dt…

Q: Find an equation of the tangent plane to the paramelric curve given by R(u, v) = ((3 + sin v)sin u,…

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Q: Find rectifying plane of et, é, t at the curve t%3DL

A: Given curve is αt=et,e-t,2t.  To find the rectifying plane of the curve . Result: Let rt→=xt,yt,zt…

Q: Give the equation of the tangent plane to each of these surfaces at the point indicated.

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Q: A curve is given by the parametrization r(t)= [²,-t, ť ]. If the tangent vector to this curve at a…

A: The given curve is,  ,

Q: Find the equation of the tangent plane at the point (-2,1 – 3) to the ellipsoid surface + y? + = 3.

A: Find the equations of the tangent plane and normal line at the point (-2, 1, -3) to the ellipsoid…

Q: x = 1+ vt, y = et X =1+1 Y. %3D

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Q: he normal vector to the tangent plane for the surface point (1, 2, -1) is: Select one: None of them…

A: topic:- application of derivatives

Q: Find a plane curve a(s) endowed with a unit-speed prametrization by SE (-0, 0) such that its signed…

A: The curvature of the curve αs is given by: αs=∫s0scos(θ(u))du, ∫s0ssin(θ(u))du-----(1) in which…

Q: The unit tangent vector of a(t) =cos 2t e, +stn 2t e, - tez at t = is

A: A vector is a quantity that has both magnitude and direction. It can be also defined as the…

Q: Sketch the curve with the vector equation ř(t) = (sin t, t). Indicate with an arrow the direction in…

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Q: Find a vector parametrization of the circle of radius 5 in the xy-plane, centered at the origin,…

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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.