Prove that the distance d from a point P (xo, Yo, 2o) to the plane r: Ar + By + Cz + D = 0 is given by |Aro + Byo + C20 + D| d = VA? + B² + C2 Use this result to compute the distance from P= (1,2, 3) to 7 : 2x - 5y- z = 9.

Q: Q1 Consider the planes. 7: 2x-y+z = 6, : x+y+22=3,, 3x-4 y+sz=10 : 2x-y +2=3,52-3y+22=1, Find the…

A: Consider the planes: π1: 2x-y+z=6, π2: x+y+2z=3, π3: 3x-4y+5z=10 and  π4: 2x-y+z=3 .   Find angles…

Q: Use Green's Theorem to calculate ( - sen(a))da + (2a + 3)dy, where C is the triangle with vertices…

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Q: Determine whether the planes 3x + 2y − z = 7, x − 4y + 2z = 0 are parallel, orthogonal, or neither.…

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Q: Determine whether the planes 5x − 3y + z = 4,  x + 4y + 7z = 1 are parallel, orthogonal, or neither.…

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Q: Show by Vector mothads that th ditonce hom fhe fornt P.(Xx, Y,, Z) to the plane Ax -+By+cz-D26 t of…

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Q: Find the distance between the parallel planes – 2x + y + z = 0 and бх — Зу — 32 -13 — 0. %3D

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Q: Find a plane through P (2, 1, 1) and perpendicular to the line of intersec- tion of the planes:…

A: find a plane through P2,1,1 and perpendicular to the line of intersection of the planes  2x+y-z=3 ,…

Q: a) Show that the distance between the parallel planes Ax + By + Cz = D1 and Ax + By + Cz = D2 is |D1…

A: (a) To show that the distance between two parallel planes Ax+By+Cz=D1 and Ax+By+Cz=D2 is…

Q: 8. Let S be the surface {(x, y, 2) | a = y? + 22 + 1}. Find every point (ro, yo, zo) on S so that…

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Q: O Find the distonce from the point pC1,\, 1) to the plane 2K-3y+72=1

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Q: (A) Find the Distance between the point (x.y.z) and the plane 3x+2y+6z = 6 If the plane meet with…

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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: Show that the plane 2x − y − 3z = 4 is parallel to the line x = −2 + 2t, y = −1 + 4t, z = 4, and…

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: A sphere of constant radius r passes through the origin O, and cuts the axes in A, B, C. Prove that…

A: Given: Equation of the sphere with origin at O x2+y2+z2-ax-by-cz=0→1The radius of the sphere is…

Q: Find the symmetrical point of point P(1, 3, 4) with respect to the plane D:3x-2y+z+6=0.

A: Symmetrical point at plane problem....

Q: Find a formula for the distance from the point P(x, y, z) to the (a) xy-plane (b) yz-plane (c)…

A: Distance between two points x1, y1, z1 and x2, y2, z2 is given by x2-x12+y2-y12+z2-z12.

Q: a) Evaluate f, (2x – z) dS, where S is part of the plane 3x + 2y + z = 1 that lies above the…

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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: Show there is no plane perpendicular to n=i+jn=i+j that passes through points P(1,2,3)P(1,2,3) and…

A: This is the rough image Point P(1,2,3) and Q(2,3,4) are the points on the plane…

Q: Find the plane containing the intersecting lines x = -1+t, y = 2+ t, z =1–t; -0 <t < o∞ | L2: x = 1…

A: Find the planes containing the intersecting lines: L1:x=-1+t,y=2+t,z=1-t for all…

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: Show that the minimum distance from the origin to a point on the plane ax + by + cz = d is |d| Ja? +…

A: Given :               ax + by + cz = d         We know that minimum distance (p) of a point x1,y1,z1…

Q: Find the distance from a point P(x1, y1, Z1) to the plane Ax + By + Cz = D.

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Q: Find for the point P(x,y,z) on the plane 3x + y – z – 15 = 0 that is closest to the origin.

A: The distance of the point P(x,y,z) from the origin is: d2=x2+y2+z2 Now we have to minimize d if the…

Q: 12. a. Use vectors to show that the distance from P;(x1, Y1, Z1) to the plane Ax + By + Cz = D is…

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Q: The pole holding up one end of a volleyball net is perpendicular to the plane containing the points…

A: Paragraph proof: You are given that VW⊥WX and VW⊥WY,which implies that ∠VWX   and   ∠ VWY are right…

Q: Evaluate x2 zdS, where S is the part of the S plane z = 1+ 2x + 3y above the rectangle [0, 2] × [0,…

A: We have to evaluate ∬Sx2zdS, where S is the part of the plane z=1+2x+3y above the rectangle 0,2×0,3.…

Q: a Find the distance from the point pC,,1) to the plane 2K -3y+77=1

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Q: a) Show that the line I = 0, y =t, :=t where teR a) lies in the plane 6r+4y-4r = 0 b) is parallel to…

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Q: Let L be a line which passes through the point (-4, 0, 1) and is perpendicular to the plane r+ 2y +…

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Q: a) Evaluate ff, (2x – z) dS, where S is part of the plane 3x + 2y + z = 1 that lies above the…

A: Surface integral

Q: (A) Find the Distance between the point (x.y.z) and the plane 3x+2y+6z = 6 8 If the plane meet with…

A: As per the company policy i can answer only first question for you, More than that leads me towards…

Q: Find m, n ∈  R so that the line r X: (n, 2,0) + λ (2, m, m) is contained in a plane π: x -3y + z = 1

A: The direction cosines of the line, r(x)= (n, 2,0) + λ (2, m, m) are &lt;2, m, m&gt; . The direction…

Q: Which of the following planes are not parallel to the plane –x + 3y + z = 3? Select all that apply.…

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Q: 02. A. Show that the plane 3x + y - z = 9 and the line {x = 2,y = -1+ 3t ,z = 1+ 3t} are parallel,…

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Q: a) Show that the line I = 0, y = t, z =t where te R a) lies in the plane 6z+4y–4x = 0 b) is parallel…

A: In the question it is asked to solve for the planes having equation given below.

Q: Prove that the planes (i) 12x – 15y + 16z – 28 = 0, (ii) 6x + 6y – 7z –8 = 0, and (üi) 2x + 35y -…

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Q: Find the distance from the point of intersection of liies ( X=t, y=3-3t, =-224) and (x=1+t y=4rt,…

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Q: Find a parametrization for the line segment beginning at P, and ending at P2. P (2,-3,2) and P2(0, –…

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Q: et L be a line which passes through the point (-3,0, 1) and is perpendicular to the plane z +…

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Q: The plane a is determined by the points A = (1,0, 1), B = (2, –1,0), and C = (1, 1, –2). L, is the…

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Q: Find the extreme values of ƒ(x, y, z) = x2yz + 1 on the intersection of the plane z = 1 with the…

A: consider the following equations  f(x,y,z)=x2yz+1                                  (1)h(x,y,z)=z-1…

Q: Prove that the perpendicular distance from the point P(xo, Yo, 20) to the plane with equation ar +…

A: Here we use the distance formula between two points.

Q: The distance from a point S(xo, Yo, Zo) to a plane M: ax + by + cz + d = 0 is given by |n PS| d =…

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Q: Let P be the plane with equation −x+2y−2z = 3. Find four distinct points on P at distance 3√10 to…

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Q: Consider the planes P and Q given as 2x-3y + z = 3 and 4x- 6y + 2z = 5: Are P and Q parallel? Find…

A: Consider the planes P and Q given as 2x-3y + z = 3 and 4x- 6y + 2z = 5: Are P and Qparallel? Find…

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: Let vector Ñ be normal to plane x – 5y – 2z = 3, and let Ē = (3, – 2, 1). Find the following. 1. The…

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Q: Describe your plan for proving the following.

A: Given that AM=MP and ∠AMP=90o. X can be thought of as a plane ground and AM is perpendicular (pole…