8. Given point P(-2,6,3) and vector B= ya, + (x+z)ay, Express P and B in cylindrical and spherical coordinates. Evaluate A at P in the Cartesian, cylindrical and spherical systems.

Q: Find an equation of the plane that passes through the point P(2, 5, 1) and has the vector n = (1,4,…

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Q: Suppose v, w, z are vectors in R". Show that if r and y are linear combinatio: v, w, and z, then r+…

A: Linear combination of the vectors

Q: The vector v with initial point (0, 0) is shown below. Write v in the form v=ai+bj, where a and b…

A: Given ,  The vector v with initial point (0, 0) We have to write v in the form v = a i + b j where…

Q: (2.3) Prove that each of the following systems a) Parabolic cylindrical coordinates (u, v, z)…

A: The given question can be solved as shown in step2.

Q: The vector E %3D —а, — 4аф + За, еxpressed in RCS and in terms of ф is E 3 (-сosф + 4sinф)ax +…

A: This question is based on unit vector conversion from Cartesian to cylindrical system.

Q: The value of the chiral vector of SWNTs if the n=8 , m=6 and the first vector length equal n/2+1 and…

A: Given that m=6,n=8to find the chiral vector of SWNT.  As we know that, the chiral vector c can be…

Q: 2 7 Consider the vectors: u = 1 v = -1 w = -2 2 Express W as a linear combination au + bv. What is…

A: follow next step

Q: A. If x and y are orthonormal vectors, then find the valne of x+ yl? - |x-yl

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Q: 2. The plane containing the point P(Xg» Yor Zo) and having the normal vector n = (a, b, c) is…

A: Equation of a plane passing through the point Px0, y0, z0 and perpendicular to the vector n=a, b, c…

Q: Given the point D(−4, 5, 3) and A = yax + (x + z)z ay, express the D and A in: Kindly answer in…

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Q: Find an equation of the plane that passes through the point P(4, 7, 1) and has the vector n = (1, 3,…

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Q: 8. Let C = 2 c=(; 1) - (;) ()- and b = Find a 2-vector z such that Cz = b.

A: we have  C=112-1b=11   we have to find a 2-vector such that  Cz=b

Q: 4. If p, 0, z are cylindrical coordinates, show that grad (log p) and grad o are solenoidal vectors.

A: In cyclindrical coordinate systemx=p Cosϕ , y=p Sinϕ , z=zP≥0 , 0≤ϕ<2π , -∞<z<∞dx=dp Cosϕ -…

Q: 2. Find the value of a for which the vector à = (axy – z³)î + (a – 2)x²j + (1 – a)xz²k have it's…

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Q: 5- The value of the chiral vector of SWNTS if the n=8, m=6 and the first vector length equal to +1…

A: n=8 m=6

Q: 11. Use vectors to show that the distance from P:(x1, yı) to the line ax + by = c is |ax, + by, – c|…

A:

Q: Let x,y,z be (non-zero) vectors and suppose that z=4x−4y and w=12x−12y−4z. Are the following…

A: A vector is a span of vectors when the first one can be written as the linear combination of the…

Q: Let the vector ü = (k,5,-1) and v=(2k,k,3) are perpendicular in R, find the value of k that related…

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Q: Which of the following vectors is coplanar to [1, 1, 1] and 2, 3]?

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Q: 2. Find the magnitude of the vector PQ whose initial point is P(1, 2) and end point is at Q(4, 3).

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Q: Show that if a E C, k E E Z, and y is parametrized by z(t) = a +re“, t E [0, 2], then |(2 - a)* dz =…

A: To Prove :-   If a ∈ ℂ , k ∈ ℤ, and γ is parametrized by z  t  = a + r eit , t ∈  0 , 2π , then ∫γ…

Q: 4. Given the two vectors rA =-ax - 3ay - 4az and re= 2ax + 2ay + 2 az and point C (1, 2, 3) find: a)…

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Q: Given point P(-2,6,3) and vector B= yax + (x+z)ay, Express P and B in cylindrical and spherical…

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Q: Show that except in degenerate cases, (u * v) * w lies in the plane of u and v, whereas u * (v * w)…

A: We will prove that  (u*v)*w=(u.w)v-(v.w)u u*v=ijkuxuyuzvxvyvz…

Q: Find a set of vectors {u, v} in R* that spans the solution set of the equations w - x -y+ 3z 0, %3D…

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Q: 3. The work Wdone by = xf + 3xyj - (x + z)k to move a particle from point 1 to point 2 is defined by…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Find the vector ū which satisfies the given conditions. i =, Ju| = 2 , ū and i have opposite…

A: The unit vector,  , in the direction of the vector is calculated as follows.

Q: Dados los vectores u=2i-3j+4k y v=(4,1,-2) , calcular la norma (módulo) del vector u - 2v V37

A: Given: u=2i-3j+4k v=4,1,-2

Q: 1. Consider the vectors a = (1, 1, 1) and y = (1, 1, 3) in Rº. Compute æ x y. 2. Show that, for æ, y…

A:

Q: Given point P(-2, 6, 3) and vector + (x + z) A =y ax + ay, express P and A in cylindrical and…

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Q: Suppose that ∇ƒ(x, y, z) is always parallel to the position vector x i + y j + z k. Show that ƒ(0,…

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Q: Find a set of vectors {u, v} in Rª that spans the solution set of the equations w – x + 2y – 4z 0,…

A: The detailed solution is given below Thank you

Q: If u = x²+2x-y and v = - 2xy - 2y, then the vorticity vector is - 3yk 10 -yk 1O IIO yk IIO

A: Given u=x2+2x-yand v=-2xy-2y

Q: Find a set of vectors {u, v} in R“ that spans the solution set of the equations 0, 0. w – x – 3z 5w…

A: To find a set of vectors {u,v} in R^4 that spans the solution set of the equations

Q: Suppose that u, v are vectors in R³, u · v = 1, || = 1, the angle between u and v is T/3, and u is…

A: According to the problem, we have

Q: Find a nonzero vector u with initial point P(-1, 3, -5) such that u has the same direction as v =…

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Q: Q3/ Show that the vectors (2i + 3j + 6k), (3i – 6j + 2k)and (6i + 2j + 3k) Are mutually…

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Q: If U and V are 2 x 1 orthonormal vectors and A = UVT then the trace of A is equal to а. О b. 2 с. -1…

A: We have to find Trace of the Matrix.

Q: 2. Given the space curve .N(77). ² B r(t) = (2 cos 2t, 2 sin 21, 31), find the unit vectors, 4

A: Consider the space curve, r→t=2cos2t , 2sin2t, 3t  To find the unit vectors T→π4 , N→π4 , B→π4 Here…

Q: Follow the method of Examples 1 and 2 to write the solution set of the given homogeneous system in…

A: We have to find the solution set of the homogeneous system…

Q: 98 25) Gvon vectors v= 30tj and w=-2i -j. Decompose v into two vectors V, and Vz, where V, is…

A:

Q: Let the vector ū = (k,5,–1) and v = (2k,k,3) are perpendicular in R', find the value of at related…

A: Given: u→=k,5,-1v→=2k,k,3

Q: 4. A unit vector orthogonal to A and whose representations are parallel to the yz-plane.

A:

Q: 2.2.12. Prove that two nonzero vectors x, y are orthogonal if and only if |x+y[? = [æ|2 + ly/².

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Q: Find an equation of the plane that passes through the point P(5, 6, 2) and has the vector n = (1,7,…

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Q: Show that the lines with vector equations L1: 7 = (-2,1,0) + s(1, –1, 1), s ER L2 : 7 = (0,1,0) +…

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Q: 1.23. Express the unit vector directed toward the point (x, Y, z) from an arbitrary point in the…

A: Express the unit vector directed toward the point (x, y, z) from an arbitrary point in the plane y =…

Q: If i • ( or v• (u x w) or w • (u x 7) equal to zero, then the 3, and w vectors u. are coplanar.…

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Q: Show that the vectors (1- i, i) and (2, –1 + i) in C“ are lincarly dependent over C but linear…

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Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: