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

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: Suppose v, w, z are vectors in R". Show that if r and y are linear combinatio: v, w, and z, then r+…

A: Linear combination of the vectors

Q: Is the line x = 9- 8t, y= 4 + 2t, z= - 5t parallel to the plane 2x + y-z=2? Give reasons for your…

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Q: Show that v = ai + bj is perpendicular to the line ax + by = c by establishing that the slope of the…

A: Given: Show that v = ai + bj is perpendicular to the line ax + by = c by establishing that the slope…

Q: Determine the value of k for which the direction vectors of the lines x+3 Z = y = -1 are…

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Q: What should k be if the vectors x=(1,1,k) y=(1,3,1) z=(k,1,3) are in the same plane in the Euclidean…

A: Three vector a1,a2,a3,b1,b2,b3 and c1,c2,c3 will be in euclidean space ℝ3 If  a1a2a3b1b2b3c1c2c3=0…

Q: Show that the lines L1 : x = 4 – t, y= 2t, z = 5+t and L2 : x = 1+4t, y= 3 – 8t, z= 5 – 4t are…

A: Let's find.

Q: Let L1 be the line passing through the points Q1(1, 1, 1) and Q2(4, −5, −8) and let L2 be the line…

A: Given L1 be the line passing through the points Q1(1, 1, 1) and Q2(4, −5, −8) L2 be the line passing…

Q: Show that the vector v = ai + bj is perpendicular to the line ax + by = c by establishing that the…

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Q: Consider these lines, where s,t, E R 11: [x, y, z] = [2, 1, -4] + t[1, 3, 1] 12: [x, y, z] = [1, k,…

A: Two lines are said to be coplanar if they lies in same plane. The equation of line passing through…

Q: 11. Use vectors to show that the distance from P:(x1, yı) to the line ax + by = c is |ax, + by, – c|…

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Q: Let the vector ü = (k,5,-1) and v=(2k,k,3) are perpendicular in R, find the value of k that related…

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Q: Q2/ Show that the vector v = ai + bj is perpendicular to the line ах + by + с.

A: A line and the vector  are said to be perpendicular to each other when the product of their sloped…

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: If v = (1,2 ) draw all vectors w = (x, y) in the xy plane with v · w = x+ 2 y = 5. Why do those w's…

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Q: 24. Consider the lines l: x = Xo +tv and m: x = x1 + su. Show that l and m intersect if and only if…

A: We will prove the given statement.

Q: Show that the lines L:a =4- t, y = 2t, z =1+t and L2:2 =1+ 21, y = 3 – 4t, z = 5 – 2t are parallel…

A: This is a problem of 3D analytical geometry and application of calculus, vector calculus.

Q: 6. Find the distance of the point (-1,1) to the line given in vector form (x, y) = (2,0) +t(4, 3).

A: Given that: The point is (-1, 1).The vector is:  (x, y)=(2, 0)+t(4, 3)

Q: d s: X = (0,1, -1) + λ (1, α, 2α) are coplanar, in which case study its relative position.

A: Consider the given lines. r: X=(1,0,2)+λ (2, 1, 3) And, s:X=(0,1, -1)+λ(1, α, 2α) Let r and s are…

Q: Determine the way in which the line [x, y, z] = [2,−3, 0] + k[−1,3,−1] intersects the plane[x, y, z]…

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Q: Let Q(0,-1,1) be a point in space and let u have the line L whose equation is L: 3- 3x = 1 – 3 y…

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Q: Suppose L1 and L2 are disjoint (nonintersecting) nonparallel lines. Is it possible for a nonzero…

A: Given:

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 the vector equation of the line, L that passes through the points P (2, -2, 8) and Q (10, 8,…

A: Given query is to find the equation of the line passing through indicated points

Q: Let L, be the line passing through the point P(-3,-5,-6) with direction vector d-[1,0,-1]", and let…

A: Line L1 passes through the point P1=(-3, -5, -6) with direction vector d1→=1, 0, -1T. ⇒L1: r→=(-3,…

Q: Show that the lines x = p + su and x = q + tv are skew lines. Find vector equations of a pair of…

A: Consider the matrix values given. The equation can be written as :

Q: 5. Which of the following is a parametrization of the line through P(-3,2, –3) and Q(1, –1,4) ? |…

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Q: Show that r|(1) and r2(1) define the same line, where rị(t) = (3, –1, 4) + t (8, 12, –6) r2(1) =…

A: Two lines have the same direction if their direction vectors v1→ and v2→ are directly proportional.…

Q: Q2/ Show that the vector v = ai + bj is perpendicular to the line ax + by + c.

A: Condition for perpendicular is: m1m2=-1 Part A: Consider the vector v=ai+bj x component of vector =…

Q: Show that the vector v = ai + bj is parallel to the line bx - ay = c

A: Find the slope of the given vector v=ai+ bj is given by ba.

Q: The equation of the plane through P.(2, 4, :5) perpendicular to the line :X = 5 +t ,Y= 1+ 3t , Z= 4t…

A: (.)  Let the point Px1,y1,z1 and perpendicular line  x-x0a=y-y0b=z-z0c Then the equation of the…

Q: 27. Are the points P(1, 2), e(7,3), and R(-2, 1) on the line with vector equation = (19,5) + s(6,1),…

A: According to our terms and conditions I have solved first question for next question repost

Q: Show that direction vectors of the lines x-3y+ 1 L: and 2 -2 are orthogonal.

A: Direction vector of a line x-x1a=y-y1b=z-z1c   is   <a,b,c>  So, Write the direction vector of…

Q: Let L, be the line in R³ which is given by the vector equation p = s Let L2 be the line in R³…

A: To find the points pair by considering the vector equation. Here, initially compute the director…

Q: In Figure 1.23, suppose that a) Z3 = 24 b) Z1 = Z4 c) Z1 = 23. Are lines r and t perpendicular?

A: Given that - ∠3 ≅ ∠4          ...........(1)∠1 ≅ ∠4       .........(2)∠1 ≅ ∠3…

Q: -) Find the vector form F(t) = ro + tu of the line through the points P(2, 7, 5) and Q(7,8, 12) Find…

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Q: Given two lines L1 : x =t+1, y = 3t + 1, z = 2t – 1, and L2 : x = 2t – 2, y = 2t + 3, z =t +1,…

A: The given lines are L1:x=t+1,y=3t+1,z=2t-1L2:x=2t-2, y=2t+3, z=t+1 and a plane H is parallel to both…

Q: Suppose T: R³ → R² is a linear map and that u and v are particular vectors in R. If T(u) = (2, –3)…

A: Given query is to find the indicated vector.

Q: Which of the following is an equation for the line through the point (2, 3) and perpendiçular to the…

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Q: Consider the lines L1 and L2 in R³ given by the vector equations L1 :ở = (2,4, 4) + t(4, 1, 5) and…

A: Given: L1 : v→ =2 , 4, 4+t 4, 1, 5 And  L2 : v→ =1, -3, 2+t -2, 3, 1

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

A: We have to solve given problems:

Q: (10) Find the point of intersection of the lines ri(t) = (1,0,0) +t(-3, 1,0) and r2(t (0, 1, 1) +…

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Q: Determine the way in which the line [x, y, z] = [2, −3, 0] + k[−1, 3, −1] intersects the plane [x,…

A: Here given as x, y, z=2, -3, 0+k-1, 3, -1⇒x=2-k⇒y=-3+3k⇒z=-k .......1 Thus the equation of the line,…

Q: Let the vector ū = (k,5,–1) and v = (2k,k,3) are perpendicular in R', find the value of at related…

A: Given: u→=k,5,-1v→=2k,k,3

Q: 1. Show that the vectors u̟ and uz parallel to lines L1 and L2 respectively, are orthogonal. L1: x =…

A: The given equations of lines are: L1:x=5-2t, y=-7+t, z=6+4t and L2:x=2+3t, y=5+2t, z=1+t Rewriting…

Q: Q1/A/ Show that in plane the non zero vector N = ai+ bj is:- perpendicular to the line ax +by +c =0…

A: We have to show that the vector N=ai+bj is perpendicular to the line ax+by+c=0. We have to find the…

Q: Show that the lines with vector equations L1: 7 = (-2,1,0) + s(1, –1, 1), s ER L2 : 7 = (0,1,0) +…

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Q: 12. Suppose that ú = (a, b) is a non-zero vector in R², where a ± 0. Show that the line L =…

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Q: Consider the vectors H [0] H . If I is the line defined by the vector w₁, fi W1 = [o] W2

A: Here, in the question three vectors are given:  w1, w2, b  L is the line defined by the vector w1 We…

Q: 2. Determine the vector equation of the plane that contains the following two lines. L1: 7 = [ 4,…

A: