Consider two lines with parametric equations p(t) = (1, −4, −3) + t(−4, 3, 0) and q(t) = (−3, 3, −1) + t(−3, 1, 0), respectively. Include a figure with your solutions. a. Find two parallel planes that contain the two lines. b. Find the distance between the lines.

Q: Find the parametric equations for the line through the point of intersection of the lines L,:x = 2…

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Q: 33 and Complete the parametric equations for the line where the planes – 3x – 9y – 12z = - 11x + 2y…

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Q: Find the distance between the skew lines with parametric equations x = 1 + t, y = 2 + 6t, z = 2t,…

A: The distance from a point P(x1,y1,z1) to a plane  is

Q: Find parametric equations for the lines in The line through the point P(3, -4, -1) parallel to the…

A: Given P3,-4,-1r=ro+tvro=3,-4,-1,ro=3i∧-4j∧-k∧v=i∧+j∧+k∧

Q: Find parametric equations for the line segment joining the points (5, 3) and (11, 11). x=t, y = 3 +…

A: Solution

Q: let L1 and L2 be the lines whose parametric equations are x-3 z-1 L1 == y - 4 = 1 =yー4= -= y – 2 4 х…

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Q: Find the parametric equations for the lines. Where, the line through the point P (3, -4, -1)…

A: Topic = Vector

Q: Express the given parametric equations of a line using bracket notation and also using i, j, k…

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Q: For the lines: L1: x=2-4t, y= 5t, z=1+2t L2: x=-7-k, y=2+3k, z= 5+k Find the parametric equations of…

A: The line passing through the point (a,b,c) and parallel to z axis is  (a,b,c)+t(0,0,1)  That is ,…

Q: The lines L1 and L2 are determined by the following symmetric and parametric equations,…

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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: Find parametric equations of the line through (-3,0, 4) that is parallel to 5i– 8j+k. x(t) y(t) =…

A: Parametric equations of a line passing through the point P(a,b,c) and parallel to the vector v→=…

Q: Consider the lines L and L2 whose symmetric equations are x – 1 L1: y + 1 z + 1 y – 3 z + 4 L2 : 4 –…

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Q: Find the distance between the skew lines with parametric equations x = 1 + t, y = 1 + 6t, z = 2t,…

A: Explanation: Given that,  Line 1: x=1+ty=1+6tz=2t Vector 1: 1,6,2 Line 2: x=1+2sy=5+15sz=-2+6s…

Q: (b) Calculate the distance from the point P(3,–1, 2) to the plane with equation 2x – y +z = 4.

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Q: Show that the curve a(t) = (t,4,²) lies in a plane and find the equation of this plane.

A: Solution:- αt=t,1+tt,1-t2t t,1t+1,1t-t=x,y,z It is clear that, t-1t-1+1t-t=x-y+z x-y+z=-1

Q: In three space, find the parametric equations of the curve C of intersection of the cylinder S: x2 +…

A: Here we have to find the parametric equations of the curve C of intersection of the cylinder S:x2+y2…

Q: Find the parametric equations of the line r = ⟨1, 2, 3⟩ + t(4, 0, -6⟩.

A: Given: r→=1,2,3+t4,0,-6 for any given vector equation of line r→=a,b,c+tp,q,r, parametric equation…

Q: Find parametric equations for the line through the points P(-1,- 1, -4) and Q(-4,-7,7). O A. x-t-3 y…

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Q: Find the point P of intersection between the line x(t) = (-3,-2, 1) + t (1, 1, -3) and the plane 4z…

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Q: Find the parametric equations for the line through the point of intersection of the lines L1:x = 2 +…

A: In line L1 and L2  't' is used as parameter but don't confuse with it. Different different line we…

Q: Find the parametric equations for the line in which the planes x-2y+4z=2 and x+2y-2z=6 intersects

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Q: Express the given parametric equations of a line using bracket notation and also using i, j. k…

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Q: Find a parametric representation of the line through the point P(3,1,-4) and perpendicular to the…

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Q: The parametric equation for the line passing through P(1,1,5) and parallel to n=(0,0,1) is defined…

A: Topic = Vector

Q: Find the distance between the skew lines with parametric equations  x = 1 + t, y = 3 + 6t, z = 2t,…

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Q: In three space, find the parametric equations of the curve C of intersection of the cylinder S: æ? +…

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Q: Find the distance between the skew lines with parametric equations x = 3 + t, y = 1 + 6t, z = 2t,…

A: I am attaching image so that you understand each and every step.

Q: Consider the lines L1 and L2 whose symmetric equations are z +1 x – 1 L1 : 2 y+ % 1 2 3 z + 4 - L2…

A: Two lines are perpendicular to each other if dot product of their direction cosine is 0.

Q: 6) Consider the plane P : x+y – z = 0 in R³ that contains both the points P(1,0, 1) and P2(0, 1, 1).…

A: • In the first part, we have to use the general Cartesian equation of a line passing through two…

Q: Find an equation of the line through (0,0,0) that is parallel to the e parametric equations x(t) = 2…

A: To find out equation of any line we need one point on the line and a vector which is parallel to the…

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

A: I have solved using an intersection point and direction ratios

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: Find parametric equations for the line segment joining the points (-6, 4) and (2, 3). x=-6 + 8t, y =…

A: Parametric equation of line segment.

Q: What is the parametric equation for the line through the point (1,2,3) parallel to the y- axis? (a)…

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Q: Find parametric equations for the line in which the planes x + 2y + z = 1 and x - y + 2z =-8…

A: x + 2y + z = 1 and x - y + 2z =-8 are the two planes.Let z=t So, the equations become:

Q: Complete the parametric equations for the line where the planes 8x + 13y – z = 27 and 9x + 15y – 12z…

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Q: Find parametric equations of the line through (-2, –1, 3) that is parallel to 2i – 3j+k. x(t) = %3D…

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Q: 2. Find the parametric and symmetric equations for the following lines. (a) r=i- 2j- 3k+t(i+j+k).…

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Q: Which of these describes a parametrized line perpendicular to the plane 3x + 2y + z = 15? a)…

A: The direction ratio of the normal vector to the plane is (3, 2, 1) So the direction ratio of the…

Q: Compute the parametric equations of the line in R3 determined by the intersection of the two planes:…

A: To obtain the parametric equations of the intersection line, use the following formulae. Let n1→…

Q: In Exercises 41–44, find a parametric equation for the line that is perpendicular to the graph of…

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Q: Let / be the line given in parametric form by the equations x = 3 +72 y = 0+32 z = -1+ 22 What is…

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Q: Find the parametric equation for the line where the planes 4x – 6y + 8z = - 36 and – 6y + z = – 15…

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Q: Find the general form of the equation of the plane passing through the point and normal to the…

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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: Complete the parametric equations for the line wherekche planes 2=0and-2z + 2y + 2z -4 intersect.…

A: The equation of the planes are: P1: 2z = 0 ⇒ z = 0                                  .........1 P2:…

Q: Find the (perpendicular) distance from the line given by the parametric equations ¤(t) -2 – 2t y(t)…

A: Solution: The objective is to find the perpendicular distance

Q: Find the parametric equations for the line of the intersection L of the two planes. 6x – 3y+ z = 5…

A: Parametric equations of the line is :  x(t) = x0 + a t  y(t) = y0 + b t  z(t) = z0 + c t  Here ( x0…

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.