a) Show that the shortest distance between the point (a, b, c) and the plane Ax + By + Cz + D = 0 is aA+bB+cC +D A² + B² + C² b) Find the shortest distance between (1,2, -3) and the plane 2x – 3y+ 6z = 20.

Q: Which of the points A(9, 0, 3), B(6, 5, 6), and C(2, 9, 2) is closest to the yz-plane? Which point…

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Q: Find an equation for the plane in R³ spanned by the points (1,5,

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Q: 4] Find the equation of the plane that contains the point (a, 0, 0), (0, b, 0) and (0, 0, c).

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Q: Draw a sketch of the plane of 4x – 4y + 2z – 9 = 0.

A: Solution

Q: Find an equation of the plane that contains all the points that are equidistant from the given…

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Q: b) Show that the points A = (1, 2, 2), B = (3,4, 5) and C = (-1,0, –1) lie on a plane through the…

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Q: Find the angle of intersection of the plane 2x+3y+5z=−32x+3y+5z=−3 with the plane 3x+2y−1z=0

A: Given: - The plane is, 2x+3y+5x=-3, with the plane 3x+2y-z=0 To find: - The angle of intersection of…

Q: 6. Determine the distance from the point (2, -1, 5) to the plane 3x - 3y + 5z = 8.

A:

Q: 2) The distance between the parallel planes ax + by + cz + d1 = 0 and |d1-d2| ax+by+cz + d2 = 0 is…

A: We find required equations of plane parallel to given plane.

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

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Q: 3) Find an equation of the plane whose points are equidistance from the point (2, - 1,1) to the…

A: As per bartleby guideline for more than one question asked only first is to be answered please…

Q: Consider the planes of equations −y+z−2=0 and x−y=0. a) Show that the planes intersect. b) Find…

A: We can solve this using definition of intersection of plane

Q: Determine the general cquation of the plane 2 passing through the point P(1,-4,-5) and parallel to…

A: We have to find equation of ohm plane

Q: Let A be the point with coordinates (4, 4, 2) and let B be the point with coordinates (3, 4, 3). Let…

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Q: Find the point in which the line X = 2, y = 3 + 2t, z = -2 – 2t; meets the plane 6х + Зу - 4z -12…

A: the point at which the line meets the plane is (x,y,z) = (2,-207,277)explanation is given in below…

Q: Verify that the two planes −3x + 6y + 7z = 1 , 6x − 12y − 14z = 25 are parallel and find the…

A: Given equations of two planes are: -3x+6y+7z=1 and, 6x-12y-14z=25 Thus we can define the normal…

Q: a)What is the distance between the point A(3,1,-1) and the plane 4x + 4y +7z +2=0 ? b) Find the…

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Q: Find an equation of the plane consisting of all points that are equidistant from (2, 4, 2) and (-2,…

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Q: Find a plane through the points P1(1, 2, 3), P2(3, 2, 1) and perpendicular to the plane 4x - y + 2z…

A: The given points and the plane are,

Q: Answer the question in the picture.

A: Concept:

Q: 37 Find the equaton of a plane that contains the point (4, -3, –1) and is perpendicular to the…

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Q: Find the general form of the equation of the plane with the given characteristics. Passes through…

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Q: Q1: Formulate the equation of plane in space that contain two points (2,4,-1) and (1,3,-2) and…

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Q: Determine the Cartesian equation of a plane that passes through a point (2,1,-5) and perpendicular…

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Q: Draw a sketch of the plane of 4x – 4y + 2z – 9 = 0. -

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Q: Find an equation of the plane containing the points P(O,0,o) and Q(2,2,2) and parallel the line…

A: Given:- Equation of line x= t, y = 1,z= t  To find:- An equation of the plane containing the points…

Q: Determine if the three planes -x-5y-5z= 2, 4x -5y +4z = 19 and x +5y - z = - 20 have an intersection…

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Q: Find an equation of the plane that contains all the points that are equidistant from the given…

A: Let the point P(x,y,z) be on the required plane which is equidistant from the given points (1, 0, 2)…

Q: Which of the points A(-4, 0, -1). B(3, 1, -1), and C(2, 4, 6) is closest to the yz-plane? Which…

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Q: Find an equation for the plane consisting of all points that are equidistant from the points (1, 0,…

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Q: 4-) Find the a Cute augle between the liue joning the two Points AC42,3),B(5,1,7) , and the Plane…

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Q: Find a full equation of the plane consisting of all points that are equidistant from (0, 1, -1) and…

A: Given two points,  (0, 1, -1) and (-4, 5, -1)  To find the equation of the plane which is…

Q: 6) Find the distance between the point (1, -4, -3) and the plane 2x – 3y + 6z = - 1.

A:

Q: 2-The distance between the point (2,3,1) and the plane 5x – 2y – 4z = 0 А- 0, В- 6.7, C- V45. is

A: Here, the distance between the given plane and point is zero. Because, the point is on the plane.

Q: Find an equation for the plane consisting of all points that are equidistant from the points (-6, 3,…

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Q: Find the equation of a plane through the point (3, -2, 8) and parallel to the plane z=x+ y.

A: Given, Plane is passing through the point (3, -2, 8) and Parallel to the plane z=x+y    Equation of…

Q: Make a sketch of two parallel planes intersected by a third plane that is not parallel to the first…

A: To draw the sketch of two parallel planes intersected by a third plane that is not parallel to the…

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

A:

Q: 4. Find the equation and sketch the graph of a plane that is parallel to the yz-coordinate plane and…

A: We know that equation of plane parallel to yz-coordinate plane is given by: x=k where k is some…

Q: 16- Find the equation of the plane that contains the line = 5 - z, and the %3D point P( 0,5,2).

A:

Q: 14) Determine the distance between the parallel planes: 6z = 4y - 2x; and 9z = 1- 3x + 6y.

A: Given problem:- Determine the distance between the parallel planes: 6z = 4y - 2x and 9z = 1 - 3x +…

Q: Check that the line (1,0,1)+t(3,0,1) is parallel to the plane -2x+4y+3z=13 and calculate the…

A: For the equation of the plane , the direction vector of the normal to the plane is given by . For…

Q: 4] Find the equation of the plane that contains the point (a, 0, 0), (0, b, 0) and (0,0, c).

A: As per our company guidelines we are supposed to answer only one question kindly post other parts in…

Q: Find distance from point A(1,2,1) to the plane containing the three points B(2,2,4), C(−1, −2, −3) &…

A: Introduction: Since you have asked multiple questions, we will solve the first question for you. If…

Q: between the point and the plane. 43. (1,−2, 3); 2x − 2y + z = 4

A: Given: The point is 1,-2, 3 and the equation of the plane is 2x-2y+z=4 To find the distance between…

Q: Find a plane containing the point (2,-1,3) and the line of intersection of the planes 5x + 7y – 6z =…

A:

Q: Which of the points A(−4, 0, −1), B(3, 1, −5), and C(2, 4, 6) is closest to the yz-plane? Which…

A: Given- A(−4, 0, −1), B(3, 1, −5), and C(2, 4, 6) To do- 1.Which of the points A(−4, 0, −1), B(3, 1,…

Q: ■ Find distance from point A(1,2,1) to the plane containing the three points B(2,2,4), C(-1, -2, -3)…

A: Distance of a point from plane

Q: Find the distance of the plane passing through the points ā, b, c from the origin.

A: First we derive the vector equation of plane.    The perpendicular distance is the projection on the…

Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.