Let R be the region in xyz-space bounded by the planes z = 1, z = 5, y = x, y = x + 4, x + y – 2z = 0, and x + y – 2z = 2. Use the change of variables %3D below to evaluate (х + z) dV. x = u – v +w, y = u + v + w, Z = u Select one: О а. 80 b. 64 Ос. 88 O d. 72 е. 96

Q: Let l,,l2,lz be the lines X(t) = (1,1, 2) + t(1,2,3), X(s)= (9,8,11) + s(2,1,1) and X (r) = (0,0,0)…

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Q: 10. Find the coordinates of the point of intersection of the line L: x = -1+ 3t, y = 2 – 2t, z = 3 +…

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Q: Let P = (0,0, 0), Q = (1, –1, –2), R = (2,1, 1). Find the area of the triangle PQR. area = Let T =…

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Q: The plane z = Ax+By+C is to be "fitted" to the following points (XkYkZk): (1,-1,0), (-1,-1,1),…

A: Let Put the given points in f we have Now, we have minimize this function.

Q: 4. For what value of h is vz in Span {v, v2} and for what value of h is {v1, V2, V3} linearly…

A: v1=1-32, v2=-39-6 and v3=5-7h

Q: Show that the line [x, y, z]= [5,-1,3]+[4,-5,2] lies on the planes 3x+2y-z = 10 and 7x+4y-4z=19.

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Q: 1. Let F(x, y,z) = (x + y², y + z²,z + x²). Let C be the triangle with vertices (1,0,0), (0,1,0),…

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Q: Let f(x, y, z) = x² + 2y? + 4z2. Find the point on the plane x + y + z = 14 at which f has its…

A: Given: fx,y,z=x2+2y2+4z2and the plane equation x+y+z=14 To find the point x,y,z The Lagrange's…

Q: Let R denote the rectangle with corners (−1, −2), (4, −2), (4,1), and (-1,1), and let P denote the…

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Q: If (x, y) and (X, Y) are the coordinates of the same point referred to two sets of rectangular axes…

A: and are the co-ordinates of the same point referred to the set of rectangular axes with same origin.

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: Evaluate ▼ (x² y + x y z²) ·dr where C is the line segment from ( 1, – 1,0) to (2,1,3 Select one: a.…

A: Given integral is

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

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Q: 30 0= +시 +씨 2 |and 3x-y-z=3 -8/

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Q: The vertices of AJKL are J(-3, 4), K(3, 0), and L(-5, -4). The equation of the median JLis y = 0.…

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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: 2. Find the value of k so that the line [x, y, z] = [2, -2, 0] + t[3, 5, k] is parallel to the plane…

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Q: Show that the line x + 1 =y/−1=(z − 2)/2 is in the plane 2x + 4y + z = 0.

A: Given equation of a line, x+1=y-1=z-22 And equation of a plane, 2x+4y+z=0

Q: Evaluate v (x? v + x y z²) · dr where C is the line segment from (1, – 1,0) to (2,1,3) | Select one:…

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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: Faith observed that her product sales in a certain town is given by the function of M(u, v) = u²v²…

A: The given function is M(u,v)=u2v2 for (u,v)∈ℝ. The given region is bounded by lines passing through…

Q: Consider, 21u = 2x – 3y and 21v = -–x+y Find the value of the Jacobian a (x, y) a (u, v)

A: 21 u = 2x - 3y, 21 v = -x + y

Q: 4. Find an equation for the plane in R³ spanned by the points (1,5, 1) and (1, 4, 2).

A: The two vectors are given as v1 = 151v2 = 142 Let xyz∈span{v1 , v2} Then xyz = mv1+nv2= m151+n142=…

Q: Let L, be the line through the points (1,2,6) and (2,4,8). And let L2 be the line of intersection of…

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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: The plane z = Ax+By+C is to be "fitted" to the following points (XkYk Zk): (1,0,0). (1,0, -1),…

A: Let  S = ∑k=14 Axk+Byk+C-zk2  For minimizing S , we have  ∂S∂A = 0 ,  ∂S∂B = 0  and  ∂S∂C = 0…

Q: Calculate the dot product of x = (3, -2, 0, 3) E R and y = (1, 2, 3, 4) ER4.

A: The dot product of two vectors x=(a1,a2,a3,a4)  and  y=(b1,b2,b3,b4)in 4th-dimensional space is: x.y…