Assume that an object is moving along a parametric curve and the three vector function T (t), N(t) , and B (t) all exist at a particular point on that curve. CIRCLE the ONE statement below that MUST BE TRUE: (a) B. T=1 (b) T x B=N (B is the binormal vector.) v (t) V (c) N (t) = I0) 시 (d) N (t) always points in the direction of velocity v (t). (e) a (t) lies in the same plane as T (t) and N (t).

Q: Find the vector and parametric equations for the line through the point P(2, 2, -2) and orthogonal…

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Q: Find parametric equations for the line whose vector equation is given. (a) (x, y) = (4, –6) + t(1,…

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Q: Represent the following equation in R² by a vector-valued/parametric equations: 2x² + 2y² – 2x + 2y…

A: The given equation is: 2x2+2y2-2x+2y=7...(1)

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A: Given information: Parametric equation of curve is : x=x(t)y=y(t)z=z(t) We need to find the length…

Q: Find a vector n normal to the plane with the equation 5x - 5y – z = 10. n = (5,

A: Given equation of plane is 5x-5y-z=-10.want to find:Normal vector n to given plane.

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Q: Find a vector parametrization of the curve x = -3z² in the xz-plane. t as the parameter in your…

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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: A particle moves in a path defined by vector-valued fanction FH)= (4)i +(zt-3)j + ( 3t - 3t) K ,…

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Q: At which point P does the curve r(t) =( 2t2, 4t+1, t³ ) given in vector form have the tangent vector…

A: Given, r(t)→=2t2,4t+1,t3  tangent vector 4,4,3 we know derivative of vector valued function gives…

Q: Find both the vector equation and the parametric equations of the line through (0,0,0) that is…

A: Given vectors area. u=2,0,2w=-2,-1,0a=0,0,0 Find the vector equation of the line perpendicular to u…

Q: Find a vector tangent to the curve of intersection of the two cyclinders x2 + y? = 8 and y? + 22 = 8…

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Q: Sketch the curve represented by the vector-valued function                               r(t) = ⟨…

A: Given  r(t) = ⟨ cos t, sin t ⟩

Q: Sketch the plane curve represented by the vector-valued function and give the orientation of the…

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

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Q: Two plane curves are defined by the vector functions r1 (t) = (t, 2t – 4) and r2 (u) = (2u, u²).…

A: r1t=t,2t-4 and r2u=2u,u2 For the intersection point, t=2u   ...1u=t22t-4=u2    ...2 Convert the…

Q: Consider the vector function      r(t) =〈t^2 - t, t^2 + t, t^3〉 Find equation of three different…

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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: Determine a direction derived from the point P and in the direction of the versor vector → informed…

A: Given:  Let f be a differentiable function at the point (a, b). Then f has a directional derivative…

Q: Find the vector equation that represents the curve of intersection of the paraboloid z 2x2 + y and…

A: z=2x2+y2  & x=ey We can see that y can be represented in terms of x.

Q: Find a vector parametrization of the curve z = :-222 in the xz-plane. Use t as the parameter in your…

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Q: Find the vector parametric equation of the line that passes through the points P(6,–2, –1) and…

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Q: Represent the plane curve by a vector-valued function. x2 + y2 = 25

A: It is given that, x2 + y2 = 25. Let x=5cost and y=sint , Then 5cost2+5sint2=25cos2t+sin2t=25

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

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Q: Find a vector tangent to the curve of intersection of the two cyclinders ? + y² = 98 and y² + z² =…

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Q: Find the value of a at which the plane curve traced out by the vector function E + bt> is not…

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Q: An airplane took off from point (3, 0, 0) at an initial velocity, v(0)= 3j with an acceleration…

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Q: Find parametric equations for the line whose vector equation is given. (a) (x, y) = (2, –5) + t(1,…

A: Parametric equation of the line, <x,y,z>=<a,b,c>+t<p,q,r> , t is a scalar is given…

Q: Find the parametric equations of the tangent line to the curve represented by the vector function 7…

A: Topic  = Vector

Q: Find the vector and parametric equations for the line through the point P(-4, -1, -3) and orthogonal…

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Q: Let f (x,y,z) = xye² + yze*. From point (1,0,1), which unit vector points in the direction in which…

A: Consider the given information. fx,y,z=xyez+yzex The point is 1,0,1. The direction in which f…

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

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Q: A velocity vector of v = 3t2i − 6t3 j − 4K ft/s  is governing the movement of an object in the 3D…

A: It is given that v=3t2i-6t3j-4kfts. Also, object starts at -1,4,5 at t=0. That is, s0=-i+4j+5k.…

Q: 1. Show that A(r - ro) + B(y - yo) =0 is an equation for the line in the ry-plane through the point…

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

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Q: Give a set of parametric equations for the line L4 that lies tangent to the graph of the vector…

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Q: Sketch the plane curve represented by the vector-valued function and give the orientation of the…

A: This is a question from Vector Calculus.

Q: Find parametric equations for the line whose vector equation is (a) (x, y) = (3, –5) + t(1, –2) X =…

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Q: R(u, v) = (uv, u sin v, v cos u)

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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: Find the vector equation that represents the curve of intersection of the paraboloid z = 4x2 + y and…

A: The given curves are: z=4x2+y2     and     x=ey

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A: A vector that makes an angle of 90 degrees with the given surface is said to be perpendicular to the…

Q: Represent the following equation in R² by a vector-valued/parametric equations: 2x² + 2y² − 2x + 2y…

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Q: Sketch the plane curve represented by the vector valued function and give the orientation of the…

A: Given rt=t3i+t2j.

Q: At which point P does the curve r(t)=( 2t³, 21², 3t – 1) givenin vector form have the tangent vector…

A: According to the problem, we have

Q: The Cartesian equation of the straight line tangent to the plane curve given parametrically by the…

A: Explanation of the answer is as follows

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.