Consider the plane P :r+y-z=0 in R that contains both the points P:(1,0,1) and P:(0, 1, 1). Answer the following questions: 6.1) Write down the parametric equations of the straight line L that contains hoth P and Pz. 6.2) Use an appropriate system of linear equations to find the equation of the plane that contains the points P, and P3, and is perpendicular to P.

Q: Consider the points A=(2,7,-4) and B=(4,5,7) in R3, one way to write the parametric equation of the…

A: Given :- Consider the points A = (2, 7, - 4) and B = (4, 5, 7) in R³, To find:- one way to write the…

Q: The equation of the plane that contains the point (1, 2, 3) and the line x = 4t, y = t, z = t is…

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Q: Find the parametric equations for the line through the point of intersection of the lines L,:x = 2…

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Q: If the plane ax+14y+cz=30 through the point P(2,3,−1) is orthogonal or perpendicular to the plane…

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Q: Which of the following is the equation of a plane parallel to the cz-coordinate plane and passing…

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Q: Find the distance between the skew lines with parametric equations x = 1 + t, y = 2 + 6t, z = 2t,…

A: The distance from a point P(x1,y1,z1) to a plane  is

Q: 5. Use the equation x2 = 4y² + 16z², to answer the following questions: %3D (a) What is the trace in…

A: (Note: We are entitled to solve only one question at a time and upto 3 subparts only) Given:…

Q: Determine the parametric equation of the plane that contains the line y-3 *+2-³-² and the point…

A: Virtual find the standard form of the plane then by satisfying the parametric values we will…

Q: Consider the planes II : 3z – y+ 2z=1, Il2 : -9z + 3y – 6z = 24 and the line of parametric equations…

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Q: Find parametric equations for the line segment joining the points (5, 3) and (11, 11). x=t, y = 3 +…

A: Solution

Q: let L1 and L2 be the lines whose parametric equations are x-3 z-1 L1 == y - 4 = 1 =yー4= -= y – 2 4 х…

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Q: The equation of the plane passing through the point A(8, 10, 9) and orthogonal to the line x = 7+…

A: Given: Point A(8,10,9) Line x=7+8t , y=6+4t , z=5+9t

Q: If the plane ar + by + cz = 30 through the points P(4, 1, 2) and Q(2, 3, – 1) is orthogonal to the…

A: Given plane passing through the given point P and Q

Q: To find the equation of a plane which cuts intercepts a, b, c res- pectively on the axes of…

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Q: Consider the plane P with equation 2x+y-2z = 4 and line L with equations x =-5+2t Ly =-5+6t z = 4-4t…

A: Given equation of plane P is 2x+y−2z=4 and equations of line are x=−5+2t, y=−5+6t, z=4−4t. And given…

Q: 2. Find a parameterization of the line formed by the intersection of the following planes: P1:2x – y…

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Q: Find the distance between the skew lines with parametric equations x = 1 + t, y = 1 + 6t, z = 2t,…

A: Explanation: Given that,  Line 1: x=1+ty=1+6tz=2t Vector 1: 1,6,2 Line 2: x=1+2sy=5+15sz=-2+6s…

Q: 2. Find a parameterization of the line formed by the intersection of the following planes: P1: 2x –…

A: Line of intersection

Q: 3. Find the distance from P(1, 1, 1) to the plane II with equation r – 3y + 4z = 10.

A: Distance between point and plane is (4√26)/13.

Q: Consider the plane P with equation 2x-2y+z =4 and line L with equations x= 3+2t y =-1-5t z= 5+4t…

A: Solved below

Q: Find the parametric equations for the line through the point of intersection of the lines L1:x = 2 +…

A: In line L1 and L2  't' is used as parameter but don't confuse with it. Different different line we…

Q: Find parametric equations for the line through P0 = (3, −1, 1) perpendicular to the plane 3x + 5y −…

A: A line L in sace with direction vector d=d1,d2,d3 that asses throughthe point P=(x0,y0,z0) can be…

Q: The angle and the parametric equations for the intersect line between planes a + y = 1, and 2r+y-2z…

A: Given, x+y=1, 2x+y-2z=2 we know angle between two plane is…

Q: Complete the parametric equations for the line where the planes – 8z – 2y – 5z = 0 and - 4x + 2y –…

A: Given: The planes are -8x-2y-5z=0 and -4x+2y-9z=0 To find the parametric equation xt, yt, zt

Q: Find a parametric representation of the line through the point P(3,1,-4) and perpendicular to the…

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Q: If the line x=1+(a+2)t, y=1-bt, z=-1+(c-1)t and the plane -x+2y-323D0 are orthogonal to each other.…

A: Given: Ratio of lineA^=<a+2,-b,+c-1> Ratio of planeB^=<-1,2,-3>

Q: 6) Consider the plane P : x+y – z = 0 in K³ that contains both the points P1(1,0, 1) and P2(-1,1,0).…

A: Given : plane P : x + y - z = 0  & Points P1(1,0,1) & P2(-1,1,0) To find :  1. Equation of…

Q: Find the distance between the skew lines with parametric equations  x = 1 + t, y = 3 + 6t, z = 2t,…

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Q: Find the angle between the two planes x- 2y + z = 29, 2x + 3y – 2z = 6. And also find the parametric…

A: here we will find angle between two planes and parametric equation of plane's line of intersection,

Q: Given the curve of intersection of the x² + 4y² -z = 0 which is a paraboloid and the plane 2x - z =…

A: Solution:-

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: Consider the lines L1 and L2 whose symmetric equations are z +1 x – 1 L1 : 2 y+ % 1 2 3 z + 4 - L2…

A: Two lines are perpendicular to each other if dot product of their direction cosine is 0.

Q: 6) Consider the plane P : x+y – z = 0 in R³ that contains both the points P(1,0, 1) and P2(0, 1, 1).…

A: • In the first part, we have to use the general Cartesian equation of a line passing through two…

Q: Find the parametric equation of the line of intersection of two planes correctly: P1:x+3y+5z=0 and…

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Q: Which of these describes a parametrized line perpendicular to the plane 3x + 2y + z = 15? A) (2 +…

A: The direction ratio of the normal vector to the plane is (3, 2,1 ) So the perpendicular line to the…

Q: Given the planes 3.x – 6y – 2z = 15 and 2x + y – 2z = 5 anser the following. (b) Find parametric…

A: I have solved using an intersection point and direction ratios

Q: Show that the curve a(t) = (t, , ) lies in a plane and find the equation of this plane.

A: For all 't' , we have α'(t) = 1 , -1t2 ,-1 - 1t2α"(t) = 0 , 2t3 , 2t3 Now  α'(t) × α"(t) = 2t3 ,…

Q: Find the set of symmetric and parametric equations of the line passing through the point P(3, 4, 1)…

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Q: Which of these describes a parametrized line perpendicular to the plane 3x + 2y + z = 15? a)…

A: The direction ratio of the normal vector to the plane is (3, 2, 1) So the direction ratio of the…

Q: Compute the parametric equations of the line in R3 determined by the intersection of the two planes:…

A: To obtain the parametric equations of the intersection line, use the following formulae. Let n1→…

Q: 1 6. Let L be the line with vector equation +t and let II be the -4 3 plane with equation 3x – 4y +…

A: Given:     Plane with equation π = 3x - 4y + z = 18        Line with vector equation L is…

Q: Consider the plane P with equation −2x+2y−z = −5 and line L with equations x = 9−4t y = −5+2t z =…

A: Consider the plane P with equation −2x+2y−z = −5 and Line with equations x = 9−4t y = −5+2t z = 4−6t

Q: The planes 3x−4y+5z=−28 and 4x−2y+4z=−24 are not parallel, so they must intersect along a line that…

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Q: a) Prove that the lines with parametric equations: x = 1+ t, y = 1 + 6t, z = 2t and x = 1+ 2s, y = 5…

A: Given that: Line L1:  x=1+t, y=1+6t, z=2tLine L2:  x=1+2s, y=5+15s, z=-2+6s

Q: 2. Find a parameterization of the line formed by the intersection of the following planes: P1: 2x –…

A: 2)as per our guideline we have to write one question at a time kindly repost remaining questions

Q: The angle and the parametric equations for the intersect line between planes a+y = 1, and 2r +y - 2z…

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Q: Consider the plane P with equation x+2y+2z=-7 and line L with equations x=-4-2t y=4+7t z=-1+3t…

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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: (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: Find the parametric equations for the line of the intersection L of the two planes. 6x – 3y+ z = 5…

A: Parametric equations of the line is :  x(t) = x0 + a t  y(t) = y0 + b t  z(t) = z0 + c t  Here ( x0…