(a) Find the cosine of the angle between the lines L1 and L2 whose vector equations are given below: L1 : ri(t) = [1, 1, 1]+ t[1,2, 3] L2 : r5(t) = [1, 1, 1]+ t[-1,4, 2]. (b) Find the equation of the plane that contains both L1 and L2.

Q: Find the vector form of the equation of the line inR3 that passes through P = ( -1, 0, 3) and is…

A: Equation of a line passes through  x1, y1, z1 having direction ratios a, b, c is, x-x1a=y-y1b=z-z1c

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)

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Q: If a line is parallel to the vector 8i + 5j + 3k and passes through the point P = (6,2,4), then the…

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Q: Find parametric equations for the line whose vector equation is given. (a) (x, y) = (4, –6) + t(1,…

A:

Q: The length of a curve given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in the…

A: Given information: Parametric equation of curve is : x=x(t)y=y(t)z=z(t) We need to find the length…

Q: O Find the parametric equation of the line through the point (-3,1,6) parallel to the vector V 2i +…

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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: 1. Let II, be the plane passing through the point (7, 1,6), and that contains the line r(t) = (1,0,…

A: Given that π1 be the plane passing through the point (7,1,6), and that contains the line…

Q: If a line is parallel to the vector 4i + 1j+ 4k and passes through the point P = (3,7,9), then the…

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Q: Find the point A (a, a, 0) on the line y = x in the xy-plane, such that the vector AB is…

A: Solution is given below:

Q: 14. Given the vector form of a line in three-space to be [x, y, z] = [2,6, – 3] + t[4,10,6], find a.…

A:

Q: Let the solid form in the orthogonal coordinate system xyz be surrounded by points A, B, C, D and F,…

A: The given co-ordinates of vertices are A = (-4, 1, 3), B = (6, 1, 1), C = ( 9, 5, 0), D = (4, 5, 1),…

Q: Find the equation of a line in vector form that is parallel to 3x−2y+z= 4, passes through the point…

A:

Q: (a) Prove that the planes 4.x – 2y + 6 = 0 and y – 4z – 13 = 0 intersect each other. (b) Find the…

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Q: If a line is parallel to the vector 4i + 10j + 7k and passes through the point P = (7,7,4), then the…

A: Solution

Q: If possible, determine the vector, parametric,symmetrical and reduced line r that: a) passes…

A: Given the lines S1:t·1,2,3, t∈ℝ and S2:y=x+2, z=x. To find a line L passing through the point P1,1,1…

Q: Find parametric equations for the line whose vector equation is given. (а) (т, у) 3 (6, —5) + t(1,…

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Q: 10.Determine the vector equation of the plane that contains the origin and the intersection of:…

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Q: A parametric equation of the line (D) passing thrdugh A and parallel to the vector v is: x=2+3t y…

A: Given, Line D = x = 2+3ty = -1-4tz = 2  It is passing through A and parallel to the vector v  We…

Q: Find a vector normal to the plane -2x - 3y = 12 - 4z.

A: Given  a vector -2x-y=12-4z. We need to find the vector normal to the plane.

Q: Find the equation of the plane that passes through the point P(1, 0, 2) and contains the line whose…

A: Hi there, At a time we can answer only one question. So, I have answered first one (#6).

Q: Find the vector equation that represents the curve of intersection of the paraboloid z 2x2 + y and…

A: z=2x2+y2  & x=ey We can see that y can be represented in terms of x.

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Q: If a line is parallel to the vector 3i +2j+ 1k and passes through the point P = (1,3,8), then the…

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Q: Find the vector form of the equation of the line inR3 that passes through P = ( - 1, 0, 3) and is…

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Q: Consider the line l in R³ given by the parametric vector equation 3 1 -2 -1 and the plane described…

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Q: Consider the points K (−1, 0, 7) and M(1, −2, 5), the vector MN = (1, 4, −3), and the line  I with…

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Q: Find parametric equations for the line whose vector equation is given. (a) (x, y) = (2, –5) + t(1,…

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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,…

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Q: Find a vector parametric equation (t) for the line through the points P = (1, −4) and Q = (−2, −9).

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Q: find the two-dimensional vector equation of the line r(t) = a +t*m such that - r(0) has norm 8 and…

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Q: If a line is parallel to the vector li + lj+ 4k and passes through the point P = (8, 4, 8), then the…

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Q: 1. Show that A(r - ro) + B(y - yo) =0 is an equation for the line in the ry-plane through the point…

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Q: 5. Show that lines lh and lz intersect, find the point of intersection, then find a parametric…

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Q: Find parametric equations for the line whose vector equation is (a) (x, y) = (3, –5) + t(1, –2) X =…

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Q: )P=(3,2,1), then the parametric equation of the line is :   a.

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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: (b) Let r(t)-ti- +(4-t) k, then the vector equation of the line tangent to the graph of r(e) at the…

A: Given:- r(t) =t2 i -(1+t)-1j +(4-t2)k To find:- Vector Equation of tangent line passing through…

Q: Find a vector equation of the line r through D4 and perpendicular to the line s through F4 and B1.…

A: Given,D4→ = 1i ^ + 2j^ + 3k^Equation of line passing through D4→ is:x-1a= y-2b=z-3cThe above line is…

Q: Given a the vector equation r(t) (4 + 4t)i + (1 + 1t)j + (1 + 4t)k, rewrite this in terms of the…

A:

Q: Find parametric equations for the line whose vector equation is given. (а) («, у) %3D (3, —3) + t(1,…

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Q: Find the vector equation that represents the curve of intersection of the paraboloid z = 4x2 + y and…

A: The given curves are: z=4x2+y2     and     x=ey

Q: If a line is parallel to the vector 7i + 9j+ 9k and passes through the point P= (2,9, 7), then the…

A: Given query is to find the equation of the line.

Q: 14. Given the vector form of a line in three-space to be [x, y, z] = [2,6, – 3] + t[4,10,6], find a.…

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.