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$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 60.6

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Wize University Linear Algebra Textbook > Equations of Lines and Planes

Intersection of Two Lines

6 Activities

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Given the planes
π1 : 4x−3y+z=2\bcb{\pi_1 \, : \, 4x - 3y + z = 2} and π2 : x+2y+2z=0\bcb{\pi_2 \, : \, x + 2y + 2z = 0}, and the lines
L1 : {x=3t+2y=−2t+1z=t, L2 : {x=−2t+1y=t+1z=t+3, L3 : {x=t+3y=−tz=2t+1\bcb{L_1 \, : \, \left\{ \begin{matrix} x & = & 3t + 2 \\ y & = & -2t + 1 \\ z & = & t \end{matrix} \right., ~ L_2 \, : \, \left\{ \begin{matrix} x & = & -2t + 1 \\ y & = & t + 1 \\ z & = & t + 3 \end{matrix} \right., ~ L_3 \, : \, \left\{ \begin{matrix} x & = & t + 3 \\ y & = & -t \\ z & = & 2t +1 \end{matrix} \right.},
and the point P1=(1,−1,2)\bcb{P_1 = (1,-1,2)}.

Determine whether the lines L1\bcb{L_1} and L2\bcb{L_2} ever intersect. If they do, give the point of intersection. If they do not, explain why.
More Intersection of Two Lines Questions:
Practice: Intersection of lines
Practice: Intersection of lines
Determine the point(s) of intersection, if any, of the followings lines, or state whether the lines are parallel or skewed.
Practice: Intersection of lines
Practice: Intersection of lines
Determine the point(s) of intersection, if any, of the followings lines, or state whether the lines are parallel or skewed.
Consider the two lines, L_1:(1,0,0)+s(1,1,1) and L_2:(-1,2,3)+t(4,0,c).
a) For what value of c two lines intersects.
b) Find the equation of the plane containing both lines.
Consider two lines: L1:(1,0,0)+s(1,1,1) and L2:(-1,2,3)+t(4,0,c).
a) For what value of c two lines intersects.
b) Find the equation of the plane containing both lines.
Practice: Intersection of lines
Practice: Intersection of lines
Which of the following statements is/are true about these lines?
L1: [1,1,2]+𝑡[−1,2,3]L_1:\ \left[1,1,2\right]+𝑡\left[−1,2,3\right]
Which of the following statements is true about the lines
L1:x=4+ty=−1+tz=−3−2tL_1:\begin{array}{l}x=4+t\\y=-1+t\\z=-3-2t\end{array}and L2:x=1+ry=2−rz=3−2rL_2:\begin{array}{l}x=1+r\\y=2-r\\z=3-2r\end{array}
Practice: Intersection of lines
Practice: Intersection of lines
Determine the point(s) of intersection, if any, of the followings lines, or state whether the lines are parallel or skewed.
Find the value of ccif the point (a,b,c)\left(a,b,c\right)is a point of intersection between the lines
L1:x=1+ty=t+2z=3−2tL_1:\begin{array}{l}x=1+t\\y=t+2\\z=3-2t\end{array}and L2:x=−1+2ry=1+rz=4−rL_2:\begin{array}{l}x=-1+2r\\y=1+r\\z=4-r\end{array}
Line Intersections
Consider the lines:
l1:(x1,x2,x3,x4)=(0,−3,−1,2)+s(1,0,2,−1)l_1:(x_1,x_2,x_3,x_4)=(0,-3,-1,2)+s(1,0,2,-1)
l2:(x1,x2,x3,x4)=(6,−4,2,0)+t(−4,1,1,0)l_2:(x_1,x_2,x_3,x_4)=(6,-4,2,0)+t(-4,1,1,0)