DERIVE the formula (shown below) for the distance D between point, P0(x0, y0, z0) and the plane, ax + by + cz + d = 0 (show all details by the required drawings).

Q: The point of intersection between the line t with the plane 2x + 3y +5z = 31 is

A: Given line ,x-1-4=y-13=z5=tTherefore any point on the line is of the form (1-4t,1+3t,5t)Put this…

Q: Find the equation of the plane which passes through O and is parallel to z – (5x + 6y) = 3 x+ y + z…

A: see 2nd step.

Q: Find the distance, d, between the point S(4,8,4) and the plane 8x + 7y + 3z = 40. The distance, d,…

A: Given:       Point =S4,8,4        Plane 8x+7y+3z=40                  8x+7y+3z-40=0 Distance between…

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 In|…

A: From a point S(x0,y0,z0) to the plane M = ax+by+cz+d=0 Let S1=x1,y1z1 be any point in the given…

Q: Q7. What angle does the line of intersection of the plane O make with the nonnegative 2x + y - z = 0…

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Q: Write a paragraph explaining how the distance formula giving the distance between two points (x1,…

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

A:

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: Use Eq. (5) to find the point P nearest to Q = (2, 1, 2) on the plane x+y+z=l.

A: Consider the provided information, Find the point P. The normal vector to the plane is 1,1,1. And…

Q: What is the distance between the point A(2,5,1) and the plane x + 2y +2z +1=0 ? Answer:

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Q: Show that the planes given by y − 7z = 16 and 2y − 14z = 5 are parallel, and find the distance…

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Q: The smaller angle (in degrees) between the planes x + y + z =1 and 2x - y + 2z = 0 is

A: Introduction :

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

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Q: Find an equation of the form ar+by+cz = d for the plane that passes through the points A(3,2,0),…

A:

Q: O Find an equation for the plane throught (l,-l,3) parallel to the plane 3x+y+Z=7

A: 1) To find the equation of the plane passing from (1,-1,3) and parallel to the plane  3x+y+z=7

Q: Find the distance (d) from the point (– 6, – 8, – 7) to the plane 5x – 2y + 2z = 25. - d =

A:

Q: Find the distance between the parallel planes – 2x + y + z = 0 and 6х — Зу — 3г — 22 — 0. - NOTE:…

A: Find the distance between the parallel planes.

Q: Find the distance from the plane given by 6x − 2y + 3z − 1 = 0 to the point (4, 1, 3).

A: The distance from plane to a point is given by  d=Ax1+By1+Cz1+DA2+B2+C2 Where Plane is Ax+By+Cz+D=0…

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: عنوان الدرس. / الموافق Find equation of the plane that cont the line s L,:X= Sty=Ct 6+2t .....…

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Q: For what positive value of cc is the acute angle between the plane containing the points A(0,0,1),…

A: According to the given information, it is required to find the positive value of c is the acute…

Q: Explain.

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Q: Find the angle between the planes x+y=D1 and 2x+y-2z%3D2?

A: Given planes : P1: x+y-1=0, P2: 2x+y-2z-2=0 ⇒n1⇀=i+j, n2⇀=2i+j-2k

Q: Find the point p(x, y, z) on the plane 2x + y - z - 5 = 0 that is closest to the origin.

A: Given:The given plane is 2x+y-z-5=0 that is closest to the origin.To find:we have to find…

Q: The distance from the point P(1,2,3) to the plane x +y + z = 0 is

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Q: Find the trace of the plane in the given coordinate plane. Зх — 9у + 82 %3D7, yz

A:

Q: The distance from the point P(2, 1, 0) to the plane 2x + y + 2z+ 6 = 0 is 3.7 True O False

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Q: Find the distance between the parallel planes – 2x + y + z = 0 and 10г — 5у — 52 — 36 — 0. NOTE:…

A:

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: Sketch the surface in R' of the equation presented. x2 + y? - 8x = 0

A: Sketch the graph on xy-plane, i.e., when z=0

Q: the

A: We will find the distance.

Q: Find the distance from Po = (3, 9, 6) to the plane 3x + 4y – 5 = 0.

A: The given equation of plane is 3x+4y-5=0 and the point is P0=3, 9, 6. The distance between a point…

Q: Find the acute angle (in degrees) between the planes 5x + y + 4z = O and a y + 2z = – 3.

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Q: Find parametrizations for the line in which the planes 3r- 6y – 2z = 3, and 2x+y-2z = 2 intersect.

A: Given Equation of planes 3x-6y-2z=3 and 2x+y-2z=2

Q: Find the point on the plane 5x +4y+z 12 that is nearest to (2,0,1). What are the values of x, y, and…

A: The plane 5x+4y+z=12Normal vecor of the plane               <5, 4, 1>The point we seek would…

Q: Find the distance from the line x = 2 + t, y = -2+ 2t, z = 4 – t and the plane 2x – 3y – 4z = 12…

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Q: find the distance from the point to the plane. (3, 0, 10), 2x + 3y + z = 2

A: the given point is (3,0,10) the given equation of the plane is 2x+3y+z=2 we have to find the…

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

A:

Q: Find b such that B = (0, b) has hyperbolic distance of u from the point A (0,0).

A: Given: B=(0,b) has  hyperbolic distance of u from the point A=(0,0) Formula used:

Q: Find the point at which the line 7(t) +t intersects the | xz plane.

A:

Q: Find the point on the plane 3x+4y+z=8that is nearest to​ (2,0,1). What are the values of​ x, y,…

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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: Find parametrizations for the lines in which the planes x - 2y + 4z = 2, x + y - 2z = 5

A: Given the planes :- x - 2y + 4z = 2,  x + y - 2z = 5   The Arbitrarily set z =1 and solve the…

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 =…

A:

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: Find the distance, d, between the point S(9,3,5) and the plane 9x + 1y + 1z= 36. The distance, d,…

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Q: Find the point of intersection between the planes TT :x+y-z =1, n, : 2x – 3y -9z = 10 and n, : x +…

A: The three planes are, π1=x+y-z=1 π2=2x-3y-9z=10 π3=x+2y-3z=-4

Q: For what positive value of c is the acute angle between the plane containing the points A(0,0,1),…

A: Calculate the vector normal to the yz plane. Equation of yz plane:x=0Equation of vector normal to yz…

Q: Find the equation of the plane in the form z = c + mx + my containing the line in the xy-plane where…

A: Plane

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.