In Exercises 30–35 compute the indicated quantities when u = (−3, 1, −4), v = (2, 0, 5), and w =( 1, 3, 13)   Find the volume of the parallelepiped determined by the vectors u, v, and w.

Q: Use the new command from Example 3 above to find the volume of the parallelepiped formed by vectors…

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Q: Find the weighted Euclidean distance between the given vectors. = n with weights w = d(u, v) = Ex: 1

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Q: Find the projection of u onto v and find the vector component of u orthogonal to v. u = −9i − 2j −…

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Q: In Exercises 13-16, find a vector normal to the plane with the given equation.15. 3(x - 4) - 8(y -…

A: Given: The equation of the plane is: 3 (x-4)-8 (y-1)+11 z=0 We have to find the vector normal to the…

Q: 11. Find the volume of the parallelepiped determined by the vectors u = 2i + 3j - k, i + j - 2k ,…

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Q: Consider (u, v) = 0 for any vectors u, v e R4. Is this an inner product? Explain why or why not.

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Q: Let u = i−5k, v = 2i−4j + k,and w = 3i−2j + 5k. (a) u · (v × w) = ______. (b) The volume of the…

A: Given :           u = i - 5k , v = 2i - 4j + k, w = 3i - 2j + 5k

Q: In Exercises 13 and 14, find the best approximation to z by vectors of the form c V + C2V2. 2 4 -2…

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Q: Find a formula for the projection of any given vector w onto the plane u · x = 0 in R', where u is a…

A: The objective is to find the formula for the projection of any given vector w onto the plane u.x=o…

Q: Find the volume of the tetrahedron spanned by the vectors u = 13 (4,0, 8), v = (0,2, 3), and w = (0,…

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Q: In Exercises 57-62, compute the angle between the two planes, defined asthe angle 0 (between O and…

A: Explanation: To find the angle between two planes, one has to find the angle between normal…

Q: Consider (u, v) = 0 for any vectors u, v e R4. Is this an inner product? Explain why or why not. %3D

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Q: Find the projection of u onto v and find the vector component of u orthogonal to v. u = ⟨9, 7⟩, v =…

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Q: 1. Find the volume of the parallelepiped PQRS if the vectors V(PQ), V(PR), and V(PS) are…

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

Q: Determine if the given vectors are coplanar u = 2i - j - k; v = 4i + 3j + 2k; w = 6i + 7j + 5k

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Q: Let u, v, and w be three mutually perpendicular vectors in ℝ3.(a) Prove that u × (v × w) = 0.(b)…

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Q: If the volume is equal to zero , then the all vectors lie in the same plane. 0 صواب ihi O

A: We have to find either the vectors lie in same plane or not if volume is zero

Q: ) Four non-coplanar points A, B, C, D are positioned such that the line AB is perpe ular to CD, and…

A: In this question, the concept of the dot product is applied. Dot Product The product of the…

Q: Use the fact that the volume of a tetrahedron with adjacent edges given by the vectors u, v and w is…

A: To find the volume of the tetrahedron whose vertices P , Q , R and S are given .

Q: Use the vector forms of Green's Theorem to calculate (a) F•n ds and (b) F:T ds for F=y°i+x°j, where…

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Q: Find the projection of u onto v and find the vector component of u orthogonal to v.…

A: Given that, u=6,7 and v=1,4

Q: Find the volume of the parallelepiped (box) determined by u, v, and w. u V i+ 3j 4i - j+4k 2i + 4k

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Q: 29. Find the volume of the parallelepiped PQRS if the vectors V(PQ). V(PR), and V(PS) are,…

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Q: Consider the vector i 3, and let V 4 = span(ï) C Rº. %3D %3D 5

A: Given: Let the vector V→=12345 and V= span(V→)

Q: Consider the three points in R³ X1 = (-1,5,8), X2 = (1, 3, 7), X3 = (2,6, 4) and let a = x1 – X2, b…

A: Part (a) When two vectors is perpendicular then their dot product is 0.

Q: Triangles in Space

A: Area of triangle formed by P,Q,R is given by  Area=PQ ⇀×PR⇀2 UNIT VECTOR is given by  unit vector=PQ…

Q: Q. For the Spheri cal polas Coozdinates (r0,d), prove that their unit vectors mutually orthogonal?…

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Q: For two-dimensional vectors u= and v= , then the dot product is simply the scalar obtained by ủ ·…

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Q: 1. Find the volume of the parallelepiped PQRS if the vectors V(PQ),v(PR), and v(PS) respectively 31…

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Q: Find x such that the vectors xi+2j -3k, i+j ,j+3k are Coplanars (lies the Same Plane) . on

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Q: Find the volume of a parallelepiped spanned by vectors from the origin to the three points (1, 1,…

A: For any given three points A,B and C, area of parallelepiped is given by [A . (B X C)].

Q: H.W 6- Find the volume of the parallelepiped for which the given vectors are three edges. C= 2i +2j…

A: Volume of parallelepiped having a1, a2, a3, b1, b2, b3, c1, c2, c3 vectors as edges is given by…

Q: In Exercises 25 and 26, find (a) the area of the parallelogram determined by vectors u and v and (b)…

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Q: Compute the curl of the following vector V defined by V(x,y,z) = e(x²+y² +z²) y

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Q: In Exercises 15–18, a. Find the area of the triangle determined by the points P, Q, and R. b. Find a…

A: We'll answer the first question since the exact one wasn't specified . Please submit a new question…

Q: Given the 3-dimensional vectors: A = (xy)i + (2yz)j +(3xz)k B = (yz)i + (2xz)j +(3xy)k Determine the…

A: Here given vectors as A=xyi+2yzj+3xzkB=yzi+2xzj+3xyk

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: 10. Find the volume of the box generated by the vectors i, j+ k, and i+ 2j + k.

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Q: Find a set of vectors {u, v} in R* that spans the solution set of the equations w -x-y+ 3z 5w + 2x +…

A: In the question it is asked to write the vector for the given equations. w-x-y-3z=05w+2x+y+3z=0 AS…

Q: Show that the vector V = (x+ 3y)i + (y – 32) j+(x – 2z)k is solenoidal. %3D

A: Solenoidal Vector : A vector function whose divergence vanished identically is called a solenoidal…

Q: By using the definition of the dot product, which vector is orthogonal (perpendicular) to á =…

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Q: Find a set of vectors {u°, v*} in R4 that spans the solution set of the equations w-x-2y+3z=0…

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Q: If X and Y are linearly independent vectors in R", then {0,X,Y} is linearly independent.

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Q: In Exercises 1-8, write the equation of the plane with normal vector npassing through the given…

A: Given,  n = k, ( 6, 7, 2) To find  the equation of the plane with normal vector npassing through the…

Q: Find x if the vectors a =i+j+ 2k;b = xi-j+k; coplanar or linearly dependent.

A: Given,       a=i+j+2kb=xi-j+kc=3i-2j-k

Q: In Rn with inner product{x, y} = xTyderive a formula for the distance between two vectorsx = (x1, .…

A: Take two vectors x and y in an inner product space x=x1,…,xn and y=y1,…,yn then x,y=xTy

Q: verify that (u * v) . w = (v * w) . u =(w * u) . v and find the volume of the parallelepiped (box)…

A: Given that, u = i + j - 2k v = - i - k w = 2i = 4j - 2k Then we get the volume of the parallelepiped…

Q: Use the fact that the volume of a tetrahedron with adjacent edges given by the vectors u, v and w is…

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Q: Suppose that A, B, and C are thecorner points of the thin triangular plate of constant density…

A: To find the vector C to the midpoint of the side AB.