Evaluate the work done between the points P, (3,1,4) and P, (6,3,6) for the conservative field F = 8xi + 8yj + 8zk. A. W= 428 B. W= - 220 C. W= 0 D. W= 220

Q: 4.) Verify the Expansion of the triple vector product A x (B x ) = B⋅ (A·C) - C (A. B) by direct…

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Q: 32. Find the equation of a plane through the origin and per- pendicular to x – y+ z = 5 and 2x + y–…

A: Note- According to Bartleby guidelines, we are supposed to do only one question. If you want more to…

Q: Let →a=a→= and →b=b→=. Find kk so that →aa→ and →bb→ will be orthogonal (form a 90 degree angle).…

A: We Know that, Two vectors u and v are said to be orthogonal vectors if and only if u . v=0 Here, a =…

Q: of ū at the poir 01

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Q: 1. For each curve, construct dr. • Let C1 be r? + y? = 16 • Let C2 be y = 5 – 2x • Let C3 be r = 2 –…

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Q: F(x, y, z) = (z sint z sin(ar), e²y-2 X b+y

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Q: 41.) Use given grid to sketch the vector field F = (y, -x). Include vectors at all integers points…

A: Given vector field is F=12y,-x

Q: 1 -2 4 17. Let a 4 a2 -3 and b : For what 7 value(s) of h is b in the plane spanned by aj and a2?

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Q: 3. Show that B = {0, 1, 2, 6, 9} is a difference set in Zjj. What are the parameters of the SBIBD…

A: For the given question ,t he first thing to do is to find the value of λ. Then, we will proceed…

Q: Let u = 4 h = V= and w= 1 . For what value of h is w in the plane spanned by u am

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Q: 2. Let F = 3.xyz i – 4xyz j + 2xyz k . Find div F and curl F. -

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Q: Find any two non collinear vectors on the plane x-y-3z +6=0

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Q: Let v = (a) 27 t (5, 2, 5) and w= (1, 5, 0). - = 4w = (b) Let 3 (t, 1, 0). For which value(s) of t…

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Q: Let A = Q1R1 = Q2R2, where Q1 and Q2 are orthogonaland R1 and R2 are both upper triangular…

A: Since A = Q1R1 = Q2R2, we can write:   Q2^TQ1R1 = R2   Since Q1 and Q2 are orthogonal, their product…

Q: Find the value of β for which w =〈−2, β〉 is parallel to v =〈4, −3〉.

A: Two vectors are parallel if the angle between them is 0 or π. That is, the vectors are pointing in…

Q: Determine the values of y such that D(9,3,-2), E(3,4,7) and F(21,y,-20) are collinear

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Q: If F = Pi+Qj+ Rk, prove that div (curl F) = 0

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Q: Show that if PER"×" is an orthogonal projector, then P+ = P.

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Q: Which of the following is the least distance from the point (2, –1,1) to the plane x + y – z = 2?

A: Given: The plane equation is x+y-z=2and the distance from the point is 2,-1,1 To find the least…

Q: 5. For which value of x do the following vectors form a linearly dependent set in R -1 y =|x, in- х,

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Q: Assume m₁ = [4., 1.5] and m₂ = [1.5, 0.5] represent the mean vectors of pattern class 1 and class 2,…

A: To find the decision boundary between two classes, we need to find the equation of the line that…

Q: Which of the following sets DA. B. -1 THE DC. vectors are linearly independent? TELBEL [BLETE] D. 8…

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Q: The dot product of two vectors can be written in the form s+ 18 t. x1 x2 in R" is defined by * ·ý =…

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Q: Show that the vector ñ = 2î – 3j is perpendicular (orthogonal) to the line 2x – 3y – 6 = 0.

A: to Show that the  given vector is perpendicular to the  given line

Q: How does the cross product i x ì relate to the vectors ở and i geometrically? Use this to compute d…

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Q: Which of these is the correct parametrization of the line connecting (1,2,5) and (0,3,– 1)? A. x =…

A: We know that parametrization of the line connecting pointsx1,y1,z1,x2,y2,z2 is given by x=x1+tx2-x1;…

Q: Q5.Find a vector parallel to the line of the intersection of the planes 2x + y – 2z = 5 and. N, = 2î…

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Q: If F = ⟨F1, F2, F3⟩ what is div (curl(F))

A: F = ⟨F1, F2, F3⟩

Q: Let r E R and --(), v- (), -- (1). 3 u = V = W = 2 For which values r ER is the set {u, v, w}…

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Q: Let ä(t) = (5, 4, 2)t + (-6, -4,1) and (u) = (-2, 1, 2)u+ (2, -8,-6). Find the acute angle between…

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Q: 23. Let w x₁ + iy₁ and 2 = x₂ + y₂. (a) Show that the distance WZ between the points w and z is |z −…

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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: