Consider the following plane. 4x + 8y + 12z = 12 (a) Find the coordinates of three points P, Q, and R in the plane, and determine the vectors PQ and PR. P = Q = R = PQ = PR = (b) Find PQ × PR.

Q: 5) Find the equation for the plane perpendicular to the plane P and containing the line L, where P:…

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Q: Find the equation of the plane through the point (2,4,3)(2,4,3) that is perpendicular to the line…

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Q: Find a vector normal to the plane with the given equation.9x − 4y − 11z = 2

A: Our Aim is to find a vector normal to the plane with the equation given below:-9x-4y-11x=2 -(i)

Q: Use the component form to generate an equation for the plane through P1(4, 1, 5) normal to n1 = i -…

A: Recall the following fact.

Q: Determine whether the planes 3x + 2y − z = 7, x − 4y + 2z = 0 are parallel, orthogonal, or neither.…

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Q: Find the vector equation for the line of intersection of the planes 5x +y-3z= -3 and 5x + z = -1 r =…

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Q: Determine whether the planes 5x − 3y + z = 4,  x + 4y + 7z = 1 are parallel, orthogonal, or neither.…

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Q: Find the vector equation for the line of intersection of the planes x + 4y + z = -5 and x + 4z = 0…

A: Here we have to find the vector equation for the line of intersection of the planes x+4y+z = -5 and…

Q: Show that the vector v = [a, b, c]* is perpendicular to the plane P with equation ax + by + cz + d =…

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Q: Find the vector equation of the line of intersection of the planes x+y+z=1 and x+2y+2z=1.

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Q: Find the angle between the plane r+y-z=21 and the line r-13y+2%3D3-2z.

A: Angle between the plane and line

Q: Suppose that the line e is represented by r(t) = (10+ 2t, 14 + 6t, 23 + 8t) and the plane P is…

A: Please refer the attached image for complete solution. THANK YOU.

Q: Find the equation of the plane passing through the point (3, 2, 2) perpendicular to the vector…

A: Given that the plane is perpendicular to the vector n=2i+3j-k. This means the direction ratio of the…

Q: What angle does the line of intersection of the planes (x 2y + 3z 5 and 4x+y-2z-7 make with the…

A: The dot product of two vectors u→, v→ is calculated using the formula: u→·v→=u→v→cosθ, where θ is…

Q: 19. Given that A is the point (5,-1,2), I is the plane with vector equation: r. (2i + 6j + 9k) =…

A: Plane and line.

Q: Find the vector equation for the line of intersection of the planes 32 + 2y :0 ánd 3* + ,0 ) +t(6, r…

A: To find the equation of line intersection of given planes.

Q: Find an equation for the plane parallel to the liner =1- 21, y = 2 + 3t %3D 2 = 1+ 2t and that…

A: Given: Plane is parallel to the line x=1-2ty=2+3tz=1+2t Also, the plane contains the line of…

Q: Find a vector normal to the plane with the given equation. 2x + 8y + 9z = 3 n =

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Q: Find an equation of the plane that passes through the given point and is perpendicular to the given…

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Q: Q5) Answer true or false for the followings: i) in triple dot of vectors (A x B). C = (B × A).C,…

A: Cross product of the vectors

Q: Find the vector equation for the line of intersection of the planes 2x – 2y – 5z = 5 and 2a + 2z = 1

A: Given  equation of planes 2x-2y-5z=5 2x+2z=1

Q: Find constants a, b, c such thatG(u,v) = (u + av,bu + v, 2u − c)parametrizes the plane 3x − 4y + z =…

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Q: Find the equation of the plane passes through the point P(4,2,1) and perpendicular to the vector…

A: Equation of the plane passing through point P(x0,y0,z0) on the plane and perpendicular to the vector…

Q: Find the vector equation for the line of intersection of the planes 5x + 4y + 5z = 1 and 5z + z = -4…

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Q: Consider the planes given by the equations x−3y−2z=1, 4x−y+3z=8      Find a vector ' v ' parallel…

A: We have given the equation of plane-                      x−3y−2z=1,                     4x−y+3z=8.…

Q: (a) For the vectors d = (2, p, 8) and b = (q, 4, 12), determine values of p and q so that the…

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Q: Find parametric equations for the lines in The line through (2, 3, 0) perpendicular to the vectors u…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: . The line (x-2z 3, y 22) intersects the plane (x+3y-:+4=0) at the point P, Find: (a) the…

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Q: ..... Q4): A: Find the equation of the plane passing through the point p(2, –1,-1) and parallel to…

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Q: Find an equation of the plane. The plane that contains the line x = 3 + 2t, y = t, z = 9 – t and is…

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Q: b) Find the center and radius of the sphere x+y+z6x+4y-10z+25%=D0. C) Compute a ·b, axb, and…

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Q: Q3:1r the two vectors A= 31+2j+5k, and B=21- j+ 4k, they located in the x and y plane, find the…

A: To find The angle between the two vectors

Q: Use the component form to generate an equation for the plane through P1(4, 1, 5) normal to n1 = i -…

A: Concept - If a plane is passing through (x0, y0, z0) and normal vector is A, B, C.Then equation of…

Q: Q1:A-Find the equation of plane passes through (0,2,-1) parallel to the plane 3x+y+z=7? b- Find…

A: 1. A. According to question,Given plane,3x+y+z=7,Given Point 0, 2, -1

Q: The unit vector normal to the plane 2x + 3y + 6z = 7 is

A: The given plane is 2x + 3y + 6z=7. The unit normal vector to fx,y is defined as n=∇F∇F. The normal…

Q: Q/ Find the Unit Vector perpendicular to the Plane 2X+22-4Y=3

A: An equation of a plane 2X + 2Z - 4Y = 3 To Find: Unit vector perpendicular to the  given plane…

Q: Find the equation of the line of intersection of the following two planes: Plane A: 2r – 3y + 2z =…

A: Given Plane A : 2x - 3y + 2z = 5Plane B : x + 2y - z = 4To find the intersection point, Put z = 0 ,…

Q: Find the vector equation of the line of intersection of the planes x+y+z = 1 and x+2y + 2z = 1.

A: The equation of the line vector of the intersection of 2 planes can be calculated using the cross…

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

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Q: Determine whether the planes 2x − z = 1, 4x + y + 8z = 10 are parallel, orthogonal, or neither. If…

A: 2x-z=1,  4x+y+8z=10 Now comparing with a1x+b1y+c1z=d1  &  a2x+b2y+c2z=d2, we get a1=2, b1=0,…

Q: Consider the following plane. 3x + 6y + 9z = 9 (a) Find the coordinates of three points P, Q, and R…

A: Given plane               3x + 6y + 9z =9

Q: b) Find equation of the Plane through point (1, 2, 3) parallel to the vectors V1 = 2i + 3j + k and…

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Q: Find an equation of the plane through the origin that is parallel to the plane 4x – 3y + 6z + 18 =…

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Q: Find the vector equation for the line of intersection of the planes x – 5y + 2z = 1 and æ + z = 1 |…

A: To find the vector equation of the line of intersection of given planes.

Q: Find the distance between the parallel planes P and P P: 2z -y+ 3z -6 =0 P: 4z - 2y + 6z - 2 =0

A: Distance between the parallel planes ax+by+cz+d1=0 and ax+by+cz+d2=0 is…

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 vector and the cartesian equations of the plane containing the point î+ 2j – k and parallel…

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Q: x + 4y + 2z = 4, and 4x -3y +4z = 7.

A: Topic - plane and vectors