4-) Find the a Cute augle between the liue joning the two Points ACY2,3),B(5,1,7) , and the Plane determined by the three Points c.cl,o,2), D ( 5,1,6), E (3,3,1).

Q: 3. Find the Scalar eguation of the plane containing the line rt) =<!,2,-37+ -3,2, 17 and the point…

A: Given problem:- Find the scalar equation plane containing the line  r (t)= &lt;1,2,-3 &gt; + t &lt;…

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

A:

Q: Find the intersection of the plane: x - z = 6, and the line r(t) = (1,0, -,1) + t(4,9,2)

A:

Q: 1. The plane containing the point (1,-5,8) and orthogonal to the line given b x=-3+15t, y=14-t,…

A:

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…

A:

Q: Determine the equation of the plane containing the points (3, 1, 4) and (2,7, 1) and is perpen-…

A:

Q: 4. Find an equation (in point-normal form) of the plane containing the points K, M, and N. 5. Let R…

A: This is a problem of Vector Algebra.

Q: 5z' + 3z gral z² - 4 plane, is

A:

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

A:

Q: Find the distance of point A(1,0,2) to the plane 3x+y+z=9?

A: Here we have to find the distance from the point A(1,0,2) to the plane 3x+y+z-9=0

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

A: We Find the equation of plane passing through P,Q,R and then substitute the point W in the Plane…

Q: Find an equation for the plane through P(1,4, 4) that contains the line of intersection of the…

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: a) Explain fully, atleast three possible relationships between two lines in R3 [c3] b) Which of the…

A: (a) The relationship between any two lines in ℝ3 is always one of the following three: They can be…

Q: 1. How can two pencils (representing lines) be held so that a plane can be passed containing both;…

A: Please submit parts 2 and 3 separately for them to be answered. For the first part, Two lines, in…

Q: Find the distance between the line 2x - 3y + z = 6, 2y- z = 4, and the plane passing through the…

A:

Q: 3. If a plane contains two distinct points P1 and P2 then show that it contains every point on the…

A:

Q: Let M be a plane that contains the point with coordinates (-2,-2,-1) and is orthogonal to the vector…

A: Let M be a plane that contains the point with coordinates -2, -2, -1 and is orthogonal to the vector…

Q: Which of the following planes is parallel to the plane 3x – 2y + z= 3? - Oz=3x- 2y + 9 z = 6x- 4y +…

A:

Q: 1. Find an equation of the plane containing the point P(3,2,7) and having a nor- mal vector i = (2,…

A: Given: Point on the plane:  A=(3, 2, 7)Normal vector to the plane is:  n→=2, 5, 2

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

A:

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

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: 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: 5) Determine whether the plane 3x – 4y + 5z = 0 and the plane - 6x + 8y – 10z - 4 = 0 are in…

A: We find out the results by using vectors in 3D

Q: 2- A plane is defined by any three points that are in the plane. If a plane contains the points…

A:

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

A:

Q: 5. Calculate the distance between the point (3, 1, -2) and the plane 2x + y-2+1. O 10 o -10 V14 O 10…

A: The equation of the plane is  2x+y-z+1 = 0 The point is x0,y0,z0 =3,1,-2 To evaluate : The distance…

Q: 8. Determine if point A(-1, -3, 9) lies on the plane T: 8x-3y+2z=4

A: The point lies on the plane if the coordinates of the point satisfy the equation of the plane.

Q: Find an equation for the plane passing through the points Q(2, 1, –1), R(-1, 3, –1), S(3, 1, 0). 1.…

A: We know that equation of a plane is given by ax+by+cz+d=0 As it passes through (2,1,-1), we get…

Q: 6. If the plane ax + by + cz = 30 through the points P(4, 1, 2) and Q(2,3, -1) is orthogonal to the…

A:

Q: Hiwed Find equations for_thefallawing lines - The line through (3, Sz=2) Perfendiculor toX-z plane

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Q: 1. Find the equations of the planes that are exactly 4 units from the plane 2x-2y+z=6

A:

Q: 2. Find the eanation of the plane Containing the point (2,3,3)'and the line Of intersÉCtion 'of the…

A: We have to find the equation of plane containing the point (2, 3, 3) and the line of intersection of…

Q: Determine if the two planes 6x + 18y + 14z = -2 and the plane containing the points (1, –1,9), (4,…

A: First we have to find Normal vector of both planes.

Q: 5. Find a plane through the points P(1,2, 3), P2(3, 2, 1) and perpendicu- lar to the plane 4x – y+…

A:

Q: The planes 3x + 6z = 1 and 2x + 2y - z = 3 intersect in a line. Show that the planes are orthogonal.

A: Given: 3x+6z=12x+2y−z=3

Q: 2. The plane with equation: 7 = (1,2,3) + m(1,2,5) + n(1,–1,3) intersects the y- and z- axes at two…

A:

Q: 4. What is the equation of the plane through (1 ,2,3) parallel to the plane x- 2y + 3z = 9? (a) x 2y…

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

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Q: Detrmine if the two planes 6x+18y+14z= -2 and the plane contaning the points (1,-1,9), (4,-1,2) and…

A:

Q: 1. Find the plane which contains the point M (4, 5,0) and the line 2+3 = ¹ = 2²

A: To find the plane which contains the point M4, 5, 0 and the line x+35=y-4-3=z-22.

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

A:

Q: 7. If a -i+3j – k, b = 2i + 4j– 2k and c = d = a+b+ de is parallel to the yz-plane. -i+ 2j + 4k,…

A: given a=i+3j-kb=2i+4j-2kc=-i+2j+4k

Q: 4.) a. Find the point of intersection of the line through (1,-5,1) and (5,-12,6), and the plane…

A:

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: ■ 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 between the point Q (3, –2, 4) and the plane 2x – 5y + z = 10 -

A:

Q: 5) Determine whether the plane 3x – 4y + 5z = 0 and the plane – 6x + 8y – 10z 4 = 0 are in…

A:

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.