Consider the curve C parameterized by ã() = (P + 21 + 3, –2t – 4,2 – 1), - 1.4 Where, 1€ [0, +o0[. a)Find the plane that contains the vectors velocity, ,ď (0) ! and acceleration, d"(0). r. b)Verify that curve C satisfies the equation of the plane of the previous point.

Q: Consider the two parametrized paths: r(t) = (t² + 7, t + 1, 25t-1), s(t) = (8t, 2t – 2, t² – 8) What…

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Q: Find a vector function that represents the curve of intersection of the paraboloid z = 7(x^2) +…

A: We have to find vector function that represents the curve of intersection of the paraboloid z =…

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: Sketch the curve of the vector function r(t) = 3ti+ (3t + 1)j,-1<ts2, and indicate the orientation…

A: The given curve in vector form: r(t)=3ti+(3t+1)j The point of t is -1≤t≤2 To find the curve at the…

Q: In Exercises 31-36, find a parametrization of the tangent line at the point indicated. t,4t 2,2t 3),…

A: Differentiation: The line must pass through the point at t=-2 is  (t,4t+2,2t+3)=(-2,-6,-1)

Q: Consider The curve of intersection of the parabolic cylinders x = y? and z = x?. Find an equation of…

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Q: (d) Find a vector function that represents the curve of intersection of the two surfaces: the xy…

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

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Q: Find a vector n normal to the plane with the equation – (x + y+ 52) = –4. n = (-1,

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Q: Find a vector parametric equation for the ellipse that lies on the plane 5y−3x+z=9 and is inside the…

A: Given that: 5y-3x+z=9 Let x=u cos v, y=u sin v So z=9+3x-5y  =9+3u cos v-5u sin v

Q: Consider the curve C parameterized by f(t)=(t^2+2t+3,-2t-4,t^2-1),t E [0,+infinite[ a)Find the plane…

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Q: 1. Find a vector which is tangent to the curve of intersection of the surfaces z = x2 – y? and xyz +…

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

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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: Answer

Q: 2. Find the equation of the normal line and the tangent to the parametric surface defined by the…

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

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Q: Determine the vector equation that represents the curve of intersection between the cylinder 2¹² +…

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Q: Compute the equation of the plane tangent to the surface x^2+3 y^2+z^3 = 8 at the point (2,1,1).

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Q: It is known that the acceleration vector for a particle is given by A(t) = 4j + 2tk. Determine the…

A: We need to determine vector equation of curve traced.

Q: The vector equation for the curve of intersection of the surfaces x = y² and z = y in terms of the…

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Q: Find a vector function, r(t), that represents the curve of intersection of the two surfaces. The…

A: The intersecting curve of two surfaces is given as follows:

Q: Which of the following vector functions represents the curve of intersection of the cylinder x2 +y =…

A: Given Data: The equation of the curve is: x2+y2=4 ......1 The equation of the plane is: x+z=3…

Q: For time 1 > 0, the position of a particle moving in the xy-plane is given by the parametric…

A: Velocity==rate of change of distance per unit time=displacementtime=dsdt. Given that: x=4t+t2  and…

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

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Q: An object moving along a curve in the xy-plane has velocity vector v(t) =. At time t = 0, the object…

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Q: Find the Vector function that represents the curve of the intersection of the two surfaces.  The…

A: Given equations are x2+y2+4z2=4 ..... (1) and x2+z2=1 ..... (2) From equation (1) z2=(4-x2-y2)/4…

Q: Find a vector tangent to the curve of intersection of the two cyclinders x² + y? = 2 and y? + z² = 2…

A: To find a vector tangent to the curve of intersection of two cylinders        x2+y2=2 and…

Q: A velocity vector of v = 3t²i – 6t³ j – 4k L is governing the movement of an object in the 3D space.…

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Q: 2. If a particle moves in the xy-plane so that at any time t > 0, its position vector is (In(r² +…

A: Solution-

Q: Find a vector function that represents the curve of intersection of the paraboloid z=7x^2+5y^2 and…

A:

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: 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 the velocity and acceleration vectors in terms ofur and uθ . r = a(1 - cos θ) and dθ/dt = 3

A: Velocity is defined as the rate of displacement. Acceleration is known as the rate of change in…

Q: The position of a particle moving in the xy-plane is given by the vector (4t, y(2t)), where y is a…

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Q: The position vector of a particle in the xy-plane is given by for all t > 0. The acceleration…

A: position vector of a particle in the xy-plane is given by &lt; t^2+1, ln(2t+3) &gt; for all t&gt;=0

Q: For time >0, the position of a particle moving in the xy-plane is given by the parametric equations…

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Q: B-For the surface z² = x² + y² find the directional derivative at the point (√2, √2,-2) in the…

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Q: Find a vector equation for the curve of intersection of the surfaces x^2 + y^2 = 4 and z = xy.

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Q: A body is spiraling upward due to rapidly rising air on a path similar to that of a helix and having…

A: Consider the position vector.

Q: Find a vector function that represents the curve of intersection of the paraboloid z = x^2 + y^2 and…

A: Intersection of paraboloid and cylinder can be obtained by representing y in the form x and z in the…

Q: Sketch the curve of intersection of the surfaces, and find a vector equation for the curve in terms…

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Q: Q1) Find equation of plane normal to vector 3i + 2j – k and pass through point (1,1,5).

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

A: Given vector rt=2sin3t,2t,-2cos3t. Then r't=6cos3t,2,6sin3t. The given point is 0,π,-2. Now the…

Q: Find a vector function that represents the curve of intersection of the surface 4x+2y-8z^2=16 and…

A: To find a vector function that represents the curve of intersection of the surface 4x+2y-8z2=16 and…

Q: Sketch the curve of intersection of the surfaces, and find a vector equation for the curve in terms…

A: First, we will sketch the curve of the intersection of the surface. The equation of the surfaces are…

Q: The planes 5x+ 2y + 2z = 5 and 3x + 5y = -22 are not parallel, so they must intersect along a line…

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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.