a. Show that |X — х У1 - у Z1 X2 - x Z2 - z = 0 Y2 - y |Xз — х Уз — у 23 — is an equation for the plane through the three noncollinear points P,(x1, y1, Z1), P2(x2, Y2, Z2), and P3(x3, y3, Z3).

Q: Determine whether the planes are parallel, perpendicular, or neither. Зх — 15у — 12z — 2 %3D 0 (a)…

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Q: Determine whether the planes 2x - 3y + 6z = 12 and 6x + 8y + 2z = 1 are parallel, orthogonal, or…

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Q: Show that the points A = (1, 2), B = (1,0), and C = collinear and express P = (-7,0) in barycentric…

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Q: Let P(x1, y1) and Q(x2, y2) be two points in the plane. Prove that the equation of the line through…

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Q: An equation of the plane passing though the points A(-1;2;0),B(0;1;2) and C(1; -1;0) is: O…

A: It is know that equation of plane passing through points x1,y1,z1,x2,y2,z2 and x3,y3,z3 is…

Q: Find the equation of the plane passing through the three points A = (-1,0, 2), B = (2, –1, 2), C =…

A: Directly use the formula for the equation of plane when 3 points are given.

Q: Find symmetric equations for the line of intersection of the planes. z %3D 4x — у — 13, Z %3D 6х +…

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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: In Exercises 29 and 30, find the plane containing the intersecting lines. 29. L1: x = -1 + t, y = 2…

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Q: Let U = (2,1,0), V = (1,0,1), and W = (2, −1,1). Find the plane through U, V, and W, expressed in…

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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: Find the equation of the plane passing through the three points A = (-1,0, 2), B = (2, –1, 2), C =…

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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: 7. Show that the equation of the plane that has x-intercept a, y-intercept b, and z-intercept c…

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Q: Determine whether the plane x + y + z = 9 and the line x = t, y = -2t + 1, z = t + 2 are parallel,…

A: The given equation of line is in parametric form, x=ty=-2tz=t+2 Convert it into the cartesian form,…

Q: Find the equation of the plane passing through the three points A = (-1,0, 2), B = (2, –1, 2), C =…

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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: Find the equation of the plane passing through the three points A = (-1,0, 2), B = (2, –1, 2), C =…

A: The given points are A=(-1,0,2), B=(2,-1,2), C=(3,-3,-3).

Q: Let P = (-2,0, 5), Q = (3, 1, 1), and R = (0,0, 4) and define a = PÓ and b= PR. Find (d) The…

A: Given three points are: P:-2,0,5, Q:3,1,1, R:0,0,4 The position vectors for the given points are:…

Q: Determine whether or not the planes f1:   3x  –  y  + z – 2 =  0 f2:     x + 2y –  z + 1 =  0 f3:…

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Q: Find an equation of the plane through (– 1, 3, –4) that is perpendicular to the planes 8x – y + z =…

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Q: Find the equation of the plane passing through the three points A = (-1,0, 2), B = (2, –1,2), C =…

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Q: Find the symmetric equations of the L that is parallel to the two planes x+y-z= 2 and z + 3y+ z =…

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Q: Show that if v and w lie in the yz-plane, then v × w is a multipleof i.

A: Both vectors v and w lie in the y-z plane, which means they must have zero magnitude component along…

Q: Prove that the intersection line of the planes P₁: 2x -y - 2z = 0 and P₂:x-2y+z = 0 is parallel to…

A: Since you have posted multiple questions and we can answer one question. So, answered the first…

Q: If a = i+3j – k, b = 2i + 4j – 2k and c = -i + 2j +4k, find a number A such that d = a +b+ Xc is…

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Q: Find the cartesian equation of the plane determined by the two intersecting lines x-4 y+1 z-2 L1: 3…

A: Here,  v1=3i+2j+kv2=5i+3j+2k v1×v2=ijk321532=i4-3-j6-5+k9-10=i-j-k Thus, the equation of the plane…

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: 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=<1,1,-1> and a2=<2,-3,4>. Let…

Q: Let L be the line whose symmetric equations are x-1=1=z+1. The equation of the plane through the…

A: Given: Symmetric equation of line L: (x-1) = (y-1)/9 = (z+1) Point P(2,3,-1)    To find: Equation of…

Q: Show that the line through two distinct points (x1, y1) and (x2, y2) in the plane has equation: | x…

A: Solution..

Q: Which of the following are equations for the plane through the points P(1, 1, -1), Q(3, 0, 2), and…

A: First find PQ and PR , two lines on the plane

Q: What points in the xy-plane satisfy the equations and inequalities in Exercises 11–12? Draw a figure…

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Q: Determine whether the planes are parallel, perpendicular, or neither. Зх — 12у — 9г — 3 %3D 0 (a)…

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Q: Which of the following is the equation of the plane that is passing through the point P(2, 2,-2) and…

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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: Exercise 3.1.26 Show that the line through two distinct points (x1, yı) and (x2, y2) in the plane…

A: As we know that: The equation of the line passing through the points x1,y1  & x2,y2.…

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: Find the equation of the plane perpendicular to the y-axis and passing through the point (3,2,-2). O…

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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: 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: a)Find the equation of the plane in 3- dimensional space passing through the points P = (1, 2, 1), Q…

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

Q: i) x² + 25 y² – 4x + 100y +4 = 0 - ii) x² – y2 – 6x = 0.

A: i) Given equation, x2+25y2-4x+100y+4=0 Now…

Q: Find the equation of the plane passing through the three points A = (-1,0,2), B = (2,–1,2), C = (3,…

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Q: Let u = 2 and v = | 0 , and let P = span{u, v}. Find an equation for the plane P.

A: Given: Let xyz be a point in Span {u, v} so there are t, r∈ℝ such that xyz=tu+rv. So,…

Q: Determine whether the planes are parallel, perpendicular, or neither. 2x – 10y – 8z – 1 = 0 (a) | -x…

A: Given problem:- Determine whether the planes are parallel, perpendicular, or neither.   (a) 2x…

Q: Prove that three points (x1, yl), (x2, y2) and (x3, y3) are collinear ( on the same straight line)…

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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: (3) Write the equation of the plane which is perpendicular to the vector ñ² = (2₁-1₁-2) and touches…

A: To find- Write the equation of the plane which is perpendicular to the vector n→ = 2, -1, -2 and…

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.…