In Exercises 25 and 26, find (a) the area of the parallelogram determined by vectors u and v and (b) the volume of the parallelepiped determined by the vectors u, v, and w. 25. u = i + j - k, v = 2i + j + k, w =-i - 2j + 3k 26. u = i + j, v = j, w = i + j + k

Q: Show that the volume V of a parallelepiped having u, v, and w as adjacent edges is V = ∣u ∙ (v ×…

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Q: In Exercises 53 and 54, use a theorem of Pappus to find the centroid of the given triangle. Use the…

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Q: Evaluate ze dV, where E is the portion of the ball centered at the origin of radius 5, x + y+< 25…

A: Given Where E is portion of the ball centered at origin, radius 5 and lies in first octant. In…

Q: 3. Find the surface area of the surface created when the cylinder y + 2- intersects the cylinder 1²…

A: surface area of the surface given by equation z=f(x,y) A=∫∫Dfx2+fy2+1dA D is the region on xy plane

Q: 5a. Consider the right square pyramid with a horizontal plane intersecting the pyramid halfway up…

A: The horizontal plane will be parallel to the base of the pyramid.

Q: Consider the parallelepiped with adjacent edges u = 5i + 2j + k v=i+j+ 7k w = i + 9j + 5k Find the…

A: Given, u=5i+2j+k,v=i+j+7k,w=i+9j+5k.

Q: Exercise 16.6.12. Given (0,0,0), Ji (1,2, 3), T2 (6, 5, 9), Üz = (4,8, 2), write down the Cartesian…

A: The barycentric coordinates of the cartesian coordinates x1,x2,x3 is 1-x1-x2-x3,x2,x3 The first…

Q: Which of the following conditions do not have to be met to prove two solids have equal volumes using…

A: Given query is to find the condition which is not required to prove volume of solid are equal.

Q: Consider the parallelepiped with adjacent edges u = 6i + 4j + 2k; v= 2i + 2j + 4k; w = 2i + 6k + 6k…

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Q: Find the area of the surface. The part of the plane z = 9 + 2x + 5y that lies above the rectangle…

A: Given: The part of the plane z=9+2x+5y that lies above the rectangle0, 9×1, 6. To Find: Area of…

Q: 2(a) Explain centroid of a triangle and rectangle with sketch.

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Q: Find the area of the surface. The part of the plane  z = 3 + 2x + 5y  that lies above the rectangle…

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Q: Find the volume of the parallelepiped having adjacent edges u, v, and w; for u = (1,-1,0), v =…

A: Given u=1,-1,0, v=0,2,2 and w=1,3,1

Q: Consider the parallelepiped with adjacent edges u = 5i + 2j + k v = i+j+ 7k w = i + 8j + 2k %| Find…

A: The volume of the parallelepiped is 253

Q: Use Exercise 22 to find the centroid of the triangle with vertices (0,0), (a, 0) and (a, b) where…

A: To find the centroid we add the x coordinates and the y coordinates respectively and then divide by…

Q: Find the volume of the solid enclosed by the coordinate plane and the plane 2x + 3y + 6z = 6 using…

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Q: Calculate the volume of the parallelepiped spanned by u = (2, 2, 1), v = (1,0, 3), w = (0, -4, 0)

A: u→=&lt;2,2,1&gt;v→=&lt;1,0,3&gt;w→=&lt;0,-4,0&gt;Volume of…

Q: Find the volume of the solid enclosed by the paraboloid z = 4 + x2 + (y – 2)2 and the planes z = 1,…

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Q: Use a scalar triple product to find the volume of the parallelepiped that has u, v, and w as…

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Q: Find the volume of the parallelepiped spanned by u, v, and w in the Figure below.

A: u=(1,0,4) v=(1,3,1) w=(-4,2,6)

Q: Find the volume of the pyramid with base in the plane z=−5 and sides formed by the three planes y=0…

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Q: Consider the parallelepiped with adjacent edges u = 8i + 7j + k v = i+j+ 2k w = i + 8j + 5k Find the…

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Q: Find the volume of a parallelepiped whose edges are ä = 7i+5j-k, 6-j+4k and e= 4i-3j+6k %3D

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Q: Use the triple scalar product to find the volume of the parallelepiped having adjacent edges u = 4i…

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Q: Among all rectangular solids defined by the inequalities 0≤x≤a, 0≤y≤ b, 0≤z≤ 1, find the one for…

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Q: Consider the parallelepiped with adjacent edges Find the volume. V = i u = Si + 5j + k v = i + j +…

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Q: Use cylindrical coordinates to show that the volume of a sphere of radius a from which a central…

A: If a cylinder of radius b is cut from a sphere of radius a, then the volume of sphere depends on the…

Q: A region R consists of a square bounded by the lines x = −7, x = 7, y = 0, and y = −14and a half…

A: Draw the region R as follows.

Q: Consider the parallelepiped with adjacent edges u = 7i + 3j + k v = i+j+ 7k w = i+ 5j + 7k Find the…

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Q: Find the volume of the parallelepiped spanned by u = (1, 0, 4), v = (1, 2, 1) and w = (-4, 2, 9) in…

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Q: Consider the parallelepiped with adjacent edges u = 4i + 6j + k v = i +j+ 9k w = i + 7j + 6k Find…

A: Given; u=4i+6j+k , v=i+j+9k , w=i+7j+6k Volume (V)=?

Q: Find the volume of the solid enclosed by the paraboloid z = 3 + x² + (y – 2)2 and the planes z = 1,…

A: Given z=3+x2+(y-2)2 and the planes z=1, x=-2,x=2,y=0 and y=2.

Q: Verify Green's theorem Where C is the boundary of the triangle formed by x=3 , y=0 , x2= 3y

A: Verify Green's theorem Where C is the boundary of the triangle formed by x=3 , y=0 , x2= 3y

Q: H.W: Find volume of box (parallelepiped) determined by u = i+j- 2k, v = -i - k and w = 2i+ 4 j - 2k

A: Solution:-

Q: Find the area of the surface. The part of the plane  z = 4 + 3x + 2y  that lies above the rectangle…

A: Consider the part of the plane that lies above the rectangle .

Q: e of the solid by subtracting two volumes, the solid enclosed by the parabolic cylinders y = 1 − x2,…

A: Here we have to find the volume of the solid  by subtracting two volumes, the solid enclosed by the…

Q: Find the area of the cap cut from the top of the sphere x2 + y2 + z2 = 1 by the plane z = √2/2.

A: Given equation of sphere is    x2+y2+z2=1     And the equation of plane is  z=22=12 But we know that…

Q: Find the volume of the pyramid with the base in the plane z= -5 and the Sides formed by the three…

A: To find the volume of the pyramid

Q: Use the triple scalar product to find the volume of the parallelepiped having adjacent edges u = 3i…

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Q: Find the area of the surface. The part of the plane z = 4 + 3x + 2y that lies above the rectangle…

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Q: Consider the parallelepiped with adjacent edges u = 3i + 2j +k v = i+j+4k w =i+6j + 2k Find the…

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Q: As discussed by our text in 16.6, the centroid of a 2D region is at cartesian coordinates (7, 9),…

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Q: Consider the parallelepiped with adjacent edges u = 5i + 5j + k v = i+j+ 8k w = i+ 5j + 7k Find the…

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Q: Consider the parallelepiped with adjacent edges u = 5i + 7j + k v = i+j+2k w = i+ 9j + 2k Find the…

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Q: 13- Find the volume V of tetrahedron bounded by the coordinates planes and the plane z = 6 - 2x + 3y

A: As per the company guidelines we can solve first question. I hope that the given solution will help…

Q: Find the volume of the wedge cut from the cylinder x2+y2=1 by the planes z = -y and z = 0.

A: The Volume of the wedge, where D is the solid and R is the region, is calculated as,

Q: Find the surface area of the part of the plane z = 4 + 6x + 2y that lies above the rectangle [1, 4]…

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Q: Find My for a triangle whose vertices are located at (0,0), (1,0), (1,1) if the density of the…

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Q: The volume of the tetrahedron bounded by the plane * ++ = :1, {a, b, c > 0} and the coordinate a…

A: Volume of the figure will be given by integral - V = ∫∫∫dzdydxLet's find limits of z,y and x - Given…