Let L be the line segment with endpoints at (1,2,3) and (-1,1,4). A light source at is at the point P= (0,4,8). The ground is modeled by the plane z+y+z=0 and the light casts a shadow of the line segment onto the ground. Give a parametrization of the shadow.

Q: Calculate the distance from point A(1, 2, 1) to the plane of equation 3x-4y + 5z = 1.

A: To find distance of point to plane using the formulae: Given point is and plane equation is .

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

A:

Q: What is the distance from the point P=(2, -1, 4) to the plane 3x+5y-2z =20

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Q: Find the distance from the point P(1, 4, 0) to the plane through A(0, 0, 0), B(2, 0, -1), and C(2,…

A: Given points A0,0,0 , B2,0,-1 & C2,-1,0The direction ratios of AB⇒2-0,0-0,-1-0=2,0,-1Similarly…

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

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Q: Find a parametric form for the line through the point (8, –6, –3) and perpendicular to the plane 2x…

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Q: Find the distance between two planes if they are parallel x+2y+3z=6 and 2x+4y+6z=5

A: Solution:GIvenx+2y+3z=6  and 2x+4y+6z=5To find the distanceFormula:If θ=0°  then the planes are…

Q: Q2/ Find the Parametric aned standand equatior of the line Hhat passer through Points (-2,1,0)…

A: The equation of line passing through given two points is given as follows:

Q: Determine the direction angles for the line with parametric equations x = 5 + 2t, y = 12 - 8t, and z…

A: Direction angle

Q: Find the distance from the point Q = (1, -3, 3) to the plane 2x + 2y + 5z = 2.

A: The equation of the plane is 2x+2y+5z=2Here we have to find the distance between the plane and the…

Q: Find a set of parametric and symmetric equations of the line satisfying the following Passing…

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Q: Determine parametric equations of the line of intersection of the planes x+2y+3z=4 and -2x+z=0. Then…

A: Let us consider two planes ax+by+cz+d=0 and a'x+b'y+c'z+d'=0  Intersection of two planes is line…

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

A: Given :x=3+ty=2+6tz=2tx=2+2sy=4+14sz=-2+5s

Q: If (x1, yı) and (x2, y2) are distinct points in the plane, show that y 1 1 =0 X1 x2 y2 is the…

A: If (x1, y1) and (x2, y2) are distinct points in the plane, to show that  xy1x1y11x2y21 =0 is the…