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: 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: For the z-coordinate of the north pole at N(11,9,12), how did you guys get 12?

A: Your question is why maximum value of z is 12?  I show this in two ways. Geometrically Analytically

Q: Calculate v x w: v = (3,8, 8), w = (1,1,4) V X w =

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Q: Let L be the line y = 6 – 4x in R?. Find vectors a and b so that v = a+tb is a vector equation for…

A: Vector equation of line

Q: Let L be the line passing through the point P(-1, -3, -2) with direction vector d=[-1, 1, 1]T. Find…

A:

Q: Suppose that u · (vxw) = 47. Find u. (wxv). u. (wxv) = i

A: Given: u·v×w=47To find:u·w×v

Q: Question 22. 11 E 5D 7 C If x = 9, is AE parallel to BD ? BC DC A. AE is parallel to BD because = -…

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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: ~(QヘP)=P v~Q =P V ~Q ylgn ihi

A:

Q: 4.18. Given vectors u = 3j – - 6k and v = 6i – 4j – 2k, find the dot product of u and v.

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Q: Let v (k,2k,1) and w (k,-1,-8). For which values of k are v andw orthogonal?

A: Calculation:Since it is given that v and w are orthogonal, v ∙ w = 0.

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

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Q: Let x = (-2,7,-7) and y = (1,6, 6) be in R°. Compute the following: x + y = ( -5x x – y = ( -5 x +…

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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: Compute the Jacobian for the substitutions x = ρsinφcosϴ,  y = ρsinφsinϴ,  z = ρcosϴ.

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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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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: