Choose a set S of four distinct points in R3 such that aff S is the plane 2x1 + x2 – 3x3 12. Justify your wor

Q: which of the toihts PC6,L3), Q (-5,-1,4), and R(0,3,8)15 c losest to the X2-Plane? which paint lies…

A: A point in 3D has (x,y,z) coordinates. A point on XY plane has (x,y,0) coordinates i.e. z=0 A point…

Q: Find the projection of d on the plane P if a 4 and P has the equation x + y+ z = 1.

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Q: Find a plane containing the point (7,-8,8) and the line F(t) = +t

A: Given that, the point is 7,-8,8 and the line is r→t=-6,-8,-7+t6,8,4. From the given line, the point…

Q: 10. Suppose that II1 and II2 are parallel planes in R’, given by: Il : ajx + bịy + C1z = di, and II2…

A: Given two plane which are parallel  Π1:a1x+b1y+c1z-d1,  Π2:a2x+b2y+c2z-d2 and another plane is…

Q: Consider the two planes, T1: X = 4 – 2s + 3t - y = 1 +s + 2t z = 3 + 4s -t; s, t eR T2 : [x, y, z] =…

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Q: Above the paraboloid z= 6-x-y² and under the plane z= 9 for x? +y? < 4 .

A: volume by double integral using  polar coordinate

Q: Find the distance (in appropriate units) between the points P(2,−1)P(2,−1)and Q(3,4)Q(3,4)in the…

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Q: Find the intersection of the planes x + y + z = 1 and 3x − 2y +z = 5.

A: Consider two planes which are not parallel to each other and intersect. Since planes are three…

Q: Evaluate / x2 zdS, where S is the part of the plane z = 4+ 3x + y above the rectangle [0, 4] x [0,…

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Q: The planes x - 2y + 32 - 3 =0 and 3x + y - z + 5 =0 are perpendicular. Select one: True False

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Q: 4- Find The equation ofthe plane That the origin andat the poiat p(u1)ond at is parallel to theline…

A: Equation of plane

Q: Prove that the planes determined by the equations ax + by + c z = d and αx + β y + γ z = δ are…

A: Given planes, ax + by + c z = d αx + β y + γ z = δ

Q: show. that theAine in.which.the planes.X+.2y-22=-5 nod5X-2y.z=.0...jntersec.is..pavalle.to.theine.…

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Q: Indicate on an xy-plane those points (x, y) for which the statement holds. |x − 3| ≤ 5 and |y + 2|…

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Q: Which of the following points is on the xy and yz planes, but is not on the xz plane? a) (2, 0, 0)…

A: We can represent 3D coordinate system.

Q: Evaluate x² zdS, where S is the part of the plane z = 2 + 3x + y above the rectangle [0, 2] × [0, 5]

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Q: A. Show that the planes (x+y- z = 1) and ( 2x - 3y + 4z = 5 ) are neither parallel nor…

A: Let's find.

Q: If the plane F contains the points P(1, 0, 0), Q(3, 2, 0), R(0, 2, 1), L O 1 6 -6 O -2

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Q: Evaluate // x² zdS, where S is the part of the S 1 + 4x + 3y above the rectangle plane z = [о, 2] x…

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Q: In the Argand's plane what is the locus of [3(22²-5z + 3) 11_2π ong

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Q: (31) Find a plane through A(2, 1, –1) and perpendicular to the line of intersection of the planes 2x…

A: Given, 2x+y-z=3 ,x+2y+z=2 The equation of the plane passing through x1,y1,z1 is given by…

Q: The line L(t) = (4 - 3t, 13 – 3t, 4t – 10) intersects the plane 2z - (4x + 2y) = 16 at the point |…

A: We have to given  line L(t)=&lt;4−3t,13−3t,4t−10&gt; which intersects the plane 2z−(4x+2y)=16 at the…

Q: Let L be the line that passes through the point  (1,3,1 ) and orthogonal to the plane 3x-2y + 5z =…

A: In the question we have to find interacting points at which line and plane intersect.

Q: Which of the following planes is parallel to the plane 3x – 2y + z= 3? - Oz=3x- 2y + 9 z = 6x- 4y +…

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Q: Evaluate x* zdS, where S is the part of the plane z = 1+4x + 3y above the rectangle [0, 3] x [0, 5]

A: Please see the below picture for detailed solution.

Q: 2. Do the three planes x + 2y + z = 4, y - z = 1, x+ 3y= 0 have at least one commom point of…

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Q: C is counterclockwise around the rectangle with vertices (0, 0), (5, 0), (5, 3), (0, 3) (a) directly…

A: The given integral is: ∮xy dx+x2 dy C is the counterclockwise around the rectangle with vertices…

Q: 12) Find the intersection of the plane: x – z = 6, and the line r(t) = (1,0, -,1) + t(4,9,2)

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Q: Show that the planes given by 2x − 4y − 3z = 5 and −4x + 8y + 6z = 1 are parallel, and find the…

A: Solution:Given2x − 4y − 3z = 5 −4x + 8y + 6z = 1To prove :The planes are parellel and to find the…

Q: In the xy plane mark all nine of these linear combinations: c[2/1]+d[0/1] with c=0, 1 , 2 and d=0, 1…

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Q: Find and describe the intersection of the following line and plane: T:2x+3y-4z =-7 and r = (4,2,5)…

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Q: A. Show that the planes (x+ y– z = 1 ) and (2x – 3y+ 4z = 5 ) are neither parallel nor…

A: let the normal vectors for the vectors be a1, a2 thus a1=&lt;1,1,-1&gt; and a2=&lt;2,-3,4&gt;. Let…

Q: 2. Do the three planes X + 2y + z = 4, y - z = 1, x+ 3y = 0 have at least one commom point of…

A: The equation of the plane is given by the formula ax+by+cz+d=0. The system of planes can be solved…

Q: 2. Find the eanation of the plane Containing the point (2,3,3)'and the line Of intersÉCtion 'of the…

A: We have to find the equation of plane containing the point (2, 3, 3) and the line of intersection of…

Q: 5. Find a plane through the points P(1,2, 3), P2(3, 2, 1) and perpendicu- lar to the plane 4x – y+…

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Q: 9. Suppose that a : R³ → R³ by perpendicular projection onto the plane 3x – 2y + z = 0. Find ↑

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Q: Show that for every real number k the plane x - 2y + z + 3 + k(2x - y - z + 1) = 0 contains the line…

A: Show that for every real number k the plane x - 2y + z + 3 + k (2x - y - z + 1) = 0 contains the…

Q: Determine whether the planes are parallel or identical −x + 4y − z = 6, −(5/2)x + 10y − (5/2)z = 15

A: Given information: The planes are −x+4y−z=6, −52x+10y−52z=15. Calculation: The normal vectors that…

Q: Q2:Prove that the line (x-1)/2=(y+1)/3=(z-2)/(4 ) is parallel to the plane x-2y+z=6 ?

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Q: Show that the line in which the planes x + 2y – 2z = 5 and 5x – 2y - z = 0 %3D intersect is parallel…

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Q: 3 Show that the planes, x-2y+4z-13D0 and 3x-6y+8z=5, if so, find the distance * .between them…

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Q: 7. If a -i+3j – k, b = 2i + 4j– 2k and c = d = a+b+ de is parallel to the yz-plane. -i+ 2j + 4k,…

A: given a=i+3j-kb=2i+4j-2kc=-i+2j+4k

Q: Evaluate JJ (² where o is the first-octant portion of the plane 2x + y + 2z = 6 (x² + 2yz)dS

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Q: 21. Determine an equation for the line of intersection of the following planes: 3x + 2y+ 5z = 4 and…

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Q: Let P be the plane 2r + u – 7 = 2 and let 0(1.0.3) be a poir

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Q: Suppose that II1 and II2 are parallel planes in R³, given by: II : ajx + biy +cjz = d1, and II2 :…

A: If ∏1: a1x+b1y+c1z=d1 and ∏2: a2x+b2y+c2z=d2 are parallel planes, then, the cross product ∏1×∏2=0.…

Q: 1. Find the Point the Plane X + 3y +27 = 6 that on %3D is nearest to the orig in.

A: To find- Find the point on the plane x + 3y + 2z = 6 that is nearest to the origin.

Q: Given Fxy 2) = (z², xy, yz) and C is a triangle in the plane x + y + z = 2 with vertices (1,1,0),…

A: We will use basic knowledge of vector calculus.

Q: Consider in R° the following planes: p:2x+y-3z=-18. 6:x-3y+4z=4. Let ë be a plane perpendicular to…

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