3. If a plane contains two distinct points P1 and P2 then show that it contains every point on the line through Pj and P2 . 4. Show that u and v are orthogonal if and only if ||u + v||? = ||u||2 + ||v||2. (Please avoid an internal approach based on coordinates.)

Q: which one of the following pair of planes is parallel? 1. O x-4y-3z-2=0 dan 3x-9y-12z-7=0 2. O 1/4…

A: To find the pair of parallel planes

Q: A set W is defined as consisting of all the planes in R^3 that are parallel to the plane x+2y-3z=1.…

A: A set W is defined as consisting of all the planes in R3 that are parallel to the plane x+2y-3z=1.…

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

A:

Q: Vaughn Pang sketched trapezoid RUIZ on a coordinate plane. RUIZ has vertices located at R(1,4),…

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Q: If there were no understood restriction to lines in a plane in Theorem 1.4.3, the theorem would be…

A: The given statement is " In space, there is exactly one line perpendicular to a given line at any…

Q: Prove the converse to theorem 2.6.2 let l and m be  two lines. If there exists a unique point P…

A: Solution:→                  Let Assume;                        Lines are not distinct.…

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: 1 2x 3y + z = 0 and 2 +t - 1 1

A:

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: 4. (a) Show that the line l given by l: 1= 2+3t, y=1+4t, z= 5+ 2t, te R, lies in the plane II given…

A: Given problem:- Show that the line  given by   l : x = 2 + 3t , y = 1 + 4t , z=5+2t ,t in R   lies…

Q: 2. ( eMsa lan--a22 3. All 2-tuples representing points in the first quadrant of the plane including…

A:

Q: 16. Consider the points P(1, 2), Q(-2,1) and R(1,0) in the plane. Which one of the following…

A: 3 points are in the plane Which Statment is true ?

Q: If the plane contains the points P(1, 0, 0), Q(3, 2, 0), R(0, 2, 1), W(a, 1, ), then a = -6 O-2

A: Solution: The equation of plane passing through points P1,0,0, Q3,2,0, R0,2,1 is x-y+3z-1=0

Q: 4. Find a plane through Po(2, 1, –1) and perpendicular to the line of in- tersection of the planes…

A: Since you have posted multiple questions in this question which are not interlinked so I have solved…

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: show that the points AC!,l,4) B(2,2,2), coplanear (in the c(o,l,5) and D (1,0,5) are Same plane).

A:

Q: 6. Find a plane with the P (3, – 4, 5) and contains the line L:= =*

A: It is given that plane passes through (3,-4,5) and the DR's of a given line is 2,-3,5.

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: Determine the way in which the line [x, y, z] = [2,−3, 0] + k[−1,3,−1] intersects the plane[x, y, z]…

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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: Given that all three points (6, 10, 10), (11, 14, 13), and (k, 8, 6) lie on a plane that contains…

A: If three points lie on the same plane then determinate of matrix made by that points is zero.…

Q: Verify Green's theorem in the plane for M = -2y and N = x , where C is the semi circle x2 + y? = 9,…

A:

Q: A. Use the figure on the right to name each of the following: 1. three collinear points 2. three…

A: A) 1) Collinear points are the points which are on the same line. Hence 3 collinear points are A , P…

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

A:

Q: 34) Show there is no plane perpendicular to n= î +î that passes through points P(1, 2, 3) and Q(2,3,…

A: Since you have submitted multiple question, we will solve the first question for you. To get the…

Q: 2. Find the plane that is orthogonal to the line : +1 = 431 = z and contains the point where l…

A: line parallel to the vector  V=⟨a,b,c⟩ passing through the point (x1,y1,z1) is given by the…

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: 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: 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 equation of the plane that contains point P(1, -1, 2) and has the normol n= 2¿+3j+Sk.

A:

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

A:

Q: is ndicular tò the plane containing the points: A = (-1,1, 2) B = (-3, 2, 1) C = (5 – 1, 1) the

A: Given that the plane is passes through the point A, B and C. We have to find the normal vector of…

Q: 7. If a =i+ 3j – k, b= 2i+ 4j – 2k and c = -i+2j+ 4k, find a number A such that d= a+b+ Ac is…

A: Given a=i+3j-k b=2i+4j-2k c= -i+2j+4k Find the  λ such that d= a+b+λc is parallel to the yz plane…

Q: Suppose P is the plane described by the equation ax+-by + cz + d = 0. Given two points (X0, Y0, Z0)…

A: Given equation of plane ax+by+cz+d=0. The plane ax+by+cz+d=0 will divide the xyz space into two…

Q: 1 Identify the statement as true or false and justify your answer. If m is perpendicular to plane T…

A:

Q: Prove that the distance between parallel planes with equations

A: Given : Two planes with equations n·x=d1 and n·x=d2

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

A:

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: 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: 2) Find an equation of the plane that contains all points that are equidistant from the points…

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Q: 1) Find the distance from the point (1, 2, 3) (a) XZ plane to the (b) Z axis

A: In this question we will use basic of geometry i.e concept of 3D geometry. We will first plot the…

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

A:

Q: Find a plane that contains S and is perpendicular to the plane that contains the points

A:

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

A:

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

A:

Q: Find points in the Euclidean plane corresponding to the points e₁-[3,-5,-1], e2=[1,-2,2], e3 [2,1,0]…

A: By Pn , we denote the projective space of dimension n, defined as follows: Pn=Rn+1 \ {0}~, where '~'…

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

A:

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: 6.) Consider the points P(-1,2,5), Q(2, –3, 0) and (i) Write an equation of the plane containing t

A: Equation of plane: We know that, the equation of plane that passes through point (x, y, z) is given…

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.