Evaluate /1+ x² + y°dS, where S is the helicoid with vector equation S r(u, v) < u cos v, u sin v, v > 0 < u < 3, 0 < v < 27

Q: 1 Suppose a line passes through the point (1, 1 and parallel to the 2' 3 plane x + y + z = 0. Then…

A: The direction vector of the normal to the plane is (1,1,1) Given that the line is parallel to the…

Q: P that is orthogonal to the vector AB. . B e distance from the point P(-3,1,9) В(-2,3, —4).

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Q: ) Find a vector parallel to the line of intersection of the planes −4x − 2y + z = −7 and  5x +2y −…

A: (a) Given planes are −4x−2y+z=−7 and 5x+2y−5z=12 We have to find the vector v→ parallel to the line…

Q: 1 of the unit vector u =- +!= j. 2/2 2/2

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Q: Find both the parametric and vector equation of the line segment between (−2, 3) and (5, −1) where 0…

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Q: Let f(x, y) = 5y² In(4x). Then Duf(1,4) in the direction of the vector (-2,-2) is

A: The explanation of answer as follows

Q: Find a unit vector u in the direction of v. Verify that ||u|| = 1. v = (-6, 6)

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Q: 4. If the projections of vector a on x,y and z −axes are 2,1,2 units respectively, and the angle at…

A: To calculate the angle we use dot product of two vectors given by Taking two vectors as U&amp;V…

Q: Find a vector of magnitude 2 parallel to the line of intersection of the planes x + 2y + z - 1 = 0…

A: Find a vector of magnitude 2 parallel to the line of intersection of the planes x + 2y + z - 1 = 0…

Q: 2 3 Let u and v %3D . Find the length of 2u and the length of -v. - Find the distance between u and…

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Q: find a vector parametrization for the line with thegiven description. Passes through P = (4, 0, 8),…

A: The parametric equation of a line in vector for is given by the expression r→=u→+tv→. Here u→ is the…

Q: nd the flux of the vector v=⟨x,y,z⟩ along the surface of the right tetrahedron below.

A: Flux of the vector

Q: if A=i+j+X, B=zi-j+3ksfind vector normal to B and in plane of A and B- ans. $2(3i+9j+x)

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Q: Find a vector parallel to the line of intersection of the planes 5x−y+2z=4 and 4x−y−3z=0v→=  (b)…

A: Find a vector parallel to the line of intersection of the planes 5x−y+2z=4 and 4x−y−3z=0

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: 1 u (-y°i +x rmal vector to

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Q: Represent the line segment from P to Q by a vector-valued function and by a set of parametric…

A: Two vectors Px1,y1,z1 and Qx2,y2,z2 Line segment P to Q: 1) Parametric form is…

Q: Let vector u= 3,4 and vector v= 6,7. Find the magnitude of 3u-4v

A: The vectors are given by u = 3,4 v = 6,7 To evaluate : The magnitude of 3u-4v

Q: Represent the line segment from P to Q by a vector-valued function and by a set of parametric…

A: we have to represent the line segment from P to Q by a vector valued function and by a set of…

Q: orthogonal to the vector y = (2)? You may want to go through the proof of When is the vector x =…

A: Here,        x = (3, 4)T       y = (-4, y2)T x and y is orthogonal if xTy = 0.

Q: Thẻ nòrmal vector to the plane 3x + 4y – 5z – 6 = 0 is A 3i+ 4j – Sk B 3i + 4j + Sk C 3i+ 4j – 6k D…

A: It is normal vector form.

Q: Find the vector projy u. u = 3 i - j+ 3 k, V = 2 i+ j+ 2 k

A: We want to have projection vector. Projection of vector u on v is given by u.vvv⏞

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

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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: Calculate the velocity and acceleration vectors, and speed for r(t) = (sin(t), cos(3t), cos(t)) when…

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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: Show that the point 1, - 2, 0 lies on both planes 3x-2y+z=7 and x-4y-2z=9. Hence, find the vector…

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Q: Find a vector equation with parameter t for the line of intersection of the planes x + y + z = 4 and…

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Q: Consider the line L with vector equation (x, y, z) = (2,3,0) + t(1, 4, –1). Determine the point on L…

A: Foot of perpendicular is the closest point on the given line, Let M is the foot of perpendicular…

Q: Give the two unit vectors u such that the function z = g(x, y) of example 3 has zero derivatives at…

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Q: Find a vector parametrization for the line with the given description. Perpendicular to the…

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Q: Prove: For nonzero vectors ū,vER, if (dot product) ū-v=0, then ū & v are linearly independent. Hint:…

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Q: 2. If A is the position vector of the point p-(x,y,z) and A=A. And ifr is the radius of the point in…

A: We apply the cross product to find the given function and simplification needed.

Q: 4. If ricost cosh t+j sint sinh t, where i and j are constant orthogonal vectors prove that (a)…

A: Given r = i cost cosht + j sint sinht

Q: The spherical equivalent of the vector B = yi + (x + z)j located at (-2,6,3) is given by

A: The variables of the spherical coordinate system r,θ,ϕ are given as.…

Q: (2t, 1 – t, 4t) is a parametrization of the line passing through the point (0, 1,0) and also…

A: The statement : &lt;2t, 1-t, 4t&gt; is a parametrization of the line passing through the point (0,…

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: Find a nonzero vector normal to the plane z − 5(x − 1) = −2(3 — y). - -

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Q: Find a unit vector u in the direction of v. Verify that llull=1 v=(15,0)

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Q: ) orthogonal to the vector y = (2)? You may want to go through the proof of When is the vector x =…

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Q: Find a vector parametrization for the line with the given description. Perpendicular to the…

A: Given points are (0,0,8). The equation of yz-plane is given by ⇒x=0 Therefore, the normal vector is…

Q: Let u = -2i + 3j, v = 6i - j, w = -3i.Find specified vector ||u + v ||2 - ||u - v||2 or scaler.

A: Given vectors are  u = -2i + 3j, v = 6i - j, w = -3i.

Q: Let F(x, y, z) = 5x² + 2y2 + 522. For which unit vector 0 will the directional derivative DoF at (2,…

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Q: 4. (8 points) Find a vector parametrization for the intersection of the plane x +y+z = 3 and the…

A: Use polar coordinates for Parameterization

Q: -2 - are the vectors U, = %3D Ug = %3D in M22 are lineor ly depon dent linearly in dependet.

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Q: ind a vector orthogonal to both u and j+k. u=2i-j+3k

A: we have been given two vectors,  u = j+k. v=2i-j+3k   the unit vector that is orthogonal to both…

Q: Represent the line segment from P to Q by a vector-valued function and by a set of parametric…

A: Given information: The points are P(0, 0, 0), Q(5, 2, 2). Calculation: Find the vector valued…

Q: The vector u = i - 2j + 2k is positioned at a point P in space with coordinates: (x, y, 2) = (1, 2,…

A: Given:    The vector u = i-2j+2k is position at a point P in space with coordinates: (x, y, z) = (1,…