(a) Write an equation of the line L, passing through the point A(1,0,3) and perpendicular(orthogonal) to the plane P: z – 3y +: =0. (b) Find the intersection point of the line L and the plane P (given in part (a)) (c) Find the distance between the point A(1,0,3) and the plane P.

Q: Find an equation for the plane through the point (0, –4, 5) which is perpendicular to the line…

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Q: Consider the plane P with equation -2x+y-2z -6 and line L with equations x=-1-2t y 5+2t z =-7-6t…

A: Distance of (a,b,c) from the plane mx+ny+oz=p is given by (ma+nb+oc-b)/sqrt(m^2+n^2+o^2)

Q: 7. a) Find the equation of the plane passes through the point P(1,2,-1), perpendicular to…

A: We have to solve given problems:

Q: Find the equation of the plane parallel to the intersecting lines (1,2 +t, -3 – 3t) and (1 – t, 2 –…

A: Let, the direction vector of 1,2+t,-3-3t is v1=0,1,-3 and the direction vector of 1-t,2-3t,-3-2t is…

Q: 25. Find an equation of the plane through the line of intersection of the planes x – z = 1 and y +…

A:

Q: Consider the plane P with equation x-2y+2z = 3 and line L with equations x = 8+6t y =-1-2t z = 1+4t…

A: Line and plane

Q: 2. Find an equation of the plane parallel to the line

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Q: 2.) ( the plane 2x-3y+z =6, and is parallel to the line of intersection of the planes x-2y+z=4 and…

A:

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

A:

Q: Find the equation of the plane containing the point P(1, 1, –1) and the line -4+t 1- t 1+t

A:

Q: @/ Find the angle between the planes (x-+y+z=1)and (x-2y+32-1). And find the symmetric equations for…

A: Given equations of planes are x+y+z=1 and x-2y+3z=1 Here, a1=1,b1=1,c1=1 & a2=1,b2=-2,c2=3…

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: A) Find the point at which the line x = 2, y = 3+ 2t, z = -2 - 2t meets the plane 6x + 3y – 4z = -12

A: Given line is x = 2, y = 3 + 2t z = -2-2t and the plane 6x + 3y - 4z = -12

Q: 15. Find an equation of the plane through the point (-2, 4, 6) and perpendicular to the line r = t,…

A:

Q: Determine the Cartesian equation of the plane that contains the line = (7,2,-1)+ a(4,5,1) and…

A: Given problem:- Determine the Cartesian equation of the plane that contains the line = (7,2,-1) +…

Q: (A) Find the Distance between the point (x.y.z) and the plane 3x+2y+6z = 6 If the plane meet with…

A:

Q: Let L be a line which passes through the point (-3,0, 1) and is perpendicular to the plane r + 2y +…

A:

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

A:

Q: Find an equation of the plane containing the point (2,1,-1) and parallel to the plane 3x – y + 2z =…

A:

Q: 286. Find the general equation of the plane that passes through the origin and is perpendicular to…

A:

Q: 1. Let L be the line given by the equation L: and S be the plane given by the equation S: 3r - 4y +…

A: Introduction: A plane is a three-dimensional figure containing three independent variables. A plane…

Q: (b) Find an equation of the plane passing through the point A(0, 9,2) and parallel to the plane 13x…

A: Given that, the given point is A(0,9,2) the parallel line is 13x+2y-z=10. To find the equation of…

Q: Consider the plane P with equation -x+2y-2z = -3 and line L with equations x = 2 y = 9+5t z =-4-4t…

A:

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

A:

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

A:

Q: The plane 6x + 8y + 60z = 700 intersects th of the highest and lowest points or coordinates

A: Given that the plane           6x+8y+60z=700  and the cone                            z2=x2+y2

Q: Find an equation of the plane containing the point (2,1,-1) and parallel to the plane 3x – y + 2z =…

A:

Q: Find the general form of the equation of the plane containing the point Q(0, –2, 10) and the lin l1…

A: Let's find.

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: Let P be the plane with equation x-2y-2z = -11. Find four distinct points on P at distance 6√2 to…

A:

Q: Find an equation of the plane through the origin that is parallel to the plane 4x-3y+5z+13=0

A:

Q: Find the cartesian equation of the plane determined by the two intersecting lines x-4 y+1 z-2 L1: 3…

A: Here,  v1=3i+2j+kv2=5i+3j+2k v1×v2=ijk321532=i4-3-j6-5+k9-10=i-j-k Thus, the equation of the plane…

Q: Find an equation of the plane A through the point P(0,1,−1) and parallel to the plane B −x+ 2y+z=−3…

A: We will find the equation of plane a and the distance between two planes as following

Q: 3. Find the equation of the plane perpendicular to the plane x + 2y + 3z = 11 and passing through…

A: Equation of plane passing through one point (a,b,c) and normal vector n = <A,B,C>…

Q: Consider the line L1 : ¤ = 3t, y =1+t, z=4+ 5t, and the plane II1 : -3x + y – 2z= 9. a) Find the…

A:

Q: 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…

A: The equation of the straight line passing through 2 points is , , .  Here, the points are .

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

A:

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

A:

Q: Find an equation of the plane with the given characteristics. -4 The plane passes through the point…

A:

Q: The z-coordinate of the point of intersection of the plane r+2y +z= 6 and the line through the…

A:

Q: An equation of a plane P passes through the point (0,0,3) and perpendicular to the line L:x =1– 2t,…

A: Given,          An equation of a plane P passes through the point 0,0,3 and perpendicular…

Q: 1. a) Given the lines L₁ : 7 = (1,2,−4) + t(k+ 1,3k + 1, k − 3) and L₂: x=2-3s, y = 1-10s, z = 3-5s,…

A:

Q: A) Find the point at which the line x = 2, y = 3+ 2t, z = -2-2t meets the !3! plane 6x + 3y- 4z -12

A: Here we have to find the point at which the line x=2, y=3+2t, z=-2-2t meets the plane 6x+3y-4z=-12…

Q: Let P be the plane with equation x-2y+2z =-10. Find four distinct points on P at distance 6 v2 to…

A:

Q: An equation of the plane passing through the points A(-1;2; 0), B(0; 1; 2) and C(1; –1;0) is:

A:

Q: Find the equation of the plane through the origin that is parallel to the plane 4.r – 2y+ 7z +12 = 0…

A: Consider the general equation of a plane, 4x-2y+7z+12=0 the normal vector of this plane will be…

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…

A:

Q: Which of the following planes is the equation for the plane with an x-intercept at P(2,0,0), a…

A:

Q: Determine the Cartesian equation of a plane that is perpendicular to the plane x - 2y + z = 6 and…

A: Given : A plane x-2y+z=6 and which contains a line -x+2-3=y+1=z+12  To find: The Cartesian equation…

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.