The distance d of a point P to the line through points A and B is the length of the components of AP that is orthogonal to AB. So the distance from P=(4,0,-3) to the line through the points A=(2,-2,4) and B=(-1,-5,-2) is?

Q: Find a line equation through the point P(3,-1,4) and parallel to the line of intersection of the…

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Q: Find symmetric equations for the line through (1, 3, -2) and perpendicular to both u and v, where u…

A: Disclaimer: "Since you have asked multiple questions, we will solve the first question for you.  If…

Q: ■ Find the acute angle between the line joining A(1,2,3) and B(5,1,7) and the plane containing point…

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Q: The equation of the plane that passes by P(2,0, -1) and that is perpendicular to the line x-5 2-y Z…

A: The plane passes by P2,0,-1 and that is perpendicular to the line x-5-2=2-y4=z-23

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: Let L, be the line passing through the point P=(8, -6, 0) with direction vector a=-3, 3, 1]", and…

A: Given, L1 is the line passing through the point P1=8,-6,0 with the direction vector d1→=-3,3,1T. So,…

Q: The distance between the line x =-2+2t ,y=2t+1, z=2-t and the line x=3-2t ,y=-2t+1 ,z=t is

A: We are supposed to find the distance between the line x =-2+2t ,y=2t+1, z=2-t and the line x=3-2t…

Q: Find the distance between line 1, and the plane parallel to it and containing line I, where 4:…

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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: Find the distance between the given parallel planes: (a) 2x - 3y + z = 4 and 4x - 6y + 2z = 3 (b)…

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Q: Find the equation of the plane through P(4, 3, 6) and perpendicular to the line joining P(4, 3, 6)…

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Q: Find the symmetric equation of line L, that passes through the points (1, 8, 9) and (5, 9, 0).…

A: 9

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

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Q: 3) Find an equation of the plane whose points are equidistance from the point (2, - 1,1) to the…

A: As per bartleby guideline for more than one question asked only first is to be answered please…

Q: Show that the lines L1 : x =1– t, y = 2t, z = 4+t and L2 : x = 1+ 3t, y= 3 – 6t, z = 5 – 3t are…

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Q: Show that the lines whose direction cosines are given by l+m+n=0,2mn+3nl-5lm=0 are perpendicular to…

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Q: Consider the line which passes through the point P(3, -1, -1), and which is parallel to the line x =…

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Q: Find the distance from the point A(2,4,3) to the plane determined by the three points B(1,4,2),…

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Q: Find an equation of the plane through (0, –9, –1) that is perpendicular to the line x – 2 = 3t, y +…

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Q: Consider the line which passes through the point P(5, -5, 5), and which is parallel to the line x =…

A: The direction vector of the line  given by:        x=1+4ty=2+1tz=3+3t is :    r→=123+t413 So, The…

Q: Parametrize the line segment joining the points P(1, 2, 0) and Q(1, 3, -1).

A: We know that equation of line passing through the points (x1,y1,z1) and (x2,y2,z2) is

Q: The distanced of a point P to the line through points A and B is the length of the component of AP…

A: Answer

Q: If the line through the point (3.-4,-1) and parallel to the vector i+j+k. The equation of the line…

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Q: Find the vector, parametric, and symmetric equa lines: (a) the line perpendicular to x+7_y-4 with 3…

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Q: Find the distance between the given parallel planes. 2z = 2y – 6x, 3z = 1 – 9x + 3y

A: Find the equation by first converting both the equation as the parallel planes then find the answer.

Q: Consider the line which passes through the point P(1, 3, -5), and which is parallel to the line æ =…

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Q: A yacht is travelling on a course parallel to and 80 kilometers from the straight shoreline of…

A: Given,         A yacht is traveling on a course parallel to and 80 kilometers from the straight…

Q: find the distance between the Linel L) through A(I,0,-1), B (-1,)andthe Line (L2) through C…

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Q: Find an equation of the plane containing the point P(1, -2, -2) perpendicular to the line through…

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Q: Determine the equation of the plane through the points (1, 0, 3) and the line l : ¯r = (3, 4, 1) +…

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Q: Find an equation of the plane through (1,8, 5) that is perpendicular to the line x – 2 = 3t, y + 3 =…

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Q: Consider the line which passes through the point P(-5, 4, 4), and which is parallel to the line x =…

A: We will solve the problem

Q: Let L, be the line passing through the point P1=(-1, -2, -2) with direction vector d-[-3, -3, 1]",…

A: L1 is the line passing through the point P1(-1 , -2, -2) with direction vectord→ = -3, -3, 1Tand L2…

Q: Find the DC’s of the line that is perpendicular to each of the two lines whose directions are and

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Q: Show that the line joining the points P(0,1,2) and Q(3,4,8) is parallel to the line joining the…

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Q: Find the coordinates of the intersection point of the lines x_1 + 3x_2 = 3 and −2x_1 + 4x_2 = 1 in…

A: Solved below

Q: find the equation of the line through (0,2,1) that is perpendicular to both u= 4i + 3j − 5k and the…

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Q: Find a line equation through the point P(1,2,3) and parallel to the line of intersection of the…

A: Planes as:                       2x+y-z=3x+2y+z=2;   The given point as:                        (1,…

Q: Find an equation of the plane that passes through P(4, 2, – 3) and is perpendicular to the line x+1…

A: Given that the plane is passes through the point (4,2,-3) and perpendicular to line x+1-3=y-12=z2

Q: Show that the line in which the planes x + 2y – 2z = 5 and 5x – 2y - z = 0 %3D intersect is parallel…

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Q: Find an equation of the plane through (4, 1, –4) that is perpendicular to the line x – 2 = 5t, y+3 =…

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Q: Parametrize the line segment joining the points P(-3, 2, -3) and Q(1, -1, 4)

A: we have to parametrize the line segment joining the points P(-3,2,-3) and Q(1,-1,4)   as we know…

Q: Consider the line which passes through the point P(-5, 4, 4), and which is parallel to the line x =…

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Q: Find the equation of the plane passing throught (0,0,-4) and perpendicular to the line joining the…

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Q: Find an equation for the plane containing the two (parallel) lines v, = (0, 1, –4) + t(1, 3, –5) and…

A: Solution: The objective is to find an equation for the plane containing the two parallel lines

Q: Let / be the line through the point with coordinates (-2,1,0) in the direction of the vector 1.…

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Q: 4:x 3+6r, y =-2+9t,: =1-12t and L:x=-1+4t, y 3+6,-8r

A: We are given L1: x=3+6t, y=-2+9t, z=1-12t. So, L1: x-36=y+29=z-1-12. Thus, a1=6, b1=9, c1=-12. We…

Q: Prove that the lines r = (1, 2, −1) + t(2, 2, 1) and r = (−1, −2, 3) + s(4, 6, −3) intersect, and…

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