Determine if there is a plane in R 3 which contains both lines l 1 a n d l 2 where: l 1 : x = 2 t + 3 , y = t + 3 , z = 3 t + 8 and l 2 : x = 3 s − 2 , y = 3 s − 1, and z = 4s+1.

Question

Determine if there is a plane in             R        3   which contains both lines       l          1           a  n  d         l          2       where:       l          1        :  x  =  2  t  +  3  ,  y  =  t  +  3  ,  z  =  3  t  +  8 and       l          2        :  x  =  3  s  −  2  ,  y  =  3  s  −  1, and z = 4s+1.

Julianna Bell · Accepted Answer

In order for a plane to contain both lines then the lines musteither be parallel or intersect...the direction of L1 is &amp;lt;2, 1, 3&amp;gt; and the direction of L2 is &amp;lt;3, 3, 4&amp;gt;since there is no k value such that...  &amp;lt;  2  ,  1  ,  3  &amp;gt;=  k  ∗  &amp;lt;  3  ,  3  ,  4  &amp;gt;      2    3    ≠      1    3    ≠      3    4  then the lines are not parallel.determine if they intersect by equating like components (x =x, y = y, z = z).x: 2t+3=3s-2...(1)y: t+3=3s-1...(2)z: 3t+8=4s+1...(3)solving (2) for t yields    t = 3s - 4sub into (1)...    2(3s - 4) + 3 = 3s - 2s = 1sub into (2)..    t + 3 = 3(1) - 1t = -1test s = 1 and t = -1 in (3)...   3(-1) + 8 = 4(1) +1therefore the lines do intersect and thus also lie in a common plane.the normal direction of that plane is orthogonal to both lines so use cross product to find it.  &amp;lt;  2  ,  1  ,  3  &amp;gt;  ×  &amp;lt;  3  ,  3  ,  4  &amp;gt;=      |                            i                          j                          k                                      2                          1                          3                                      3                          3                          4                      |    =&amp;lt;  −  5  ,  1  ,  3  &amp;gt;we just need a point from either line to plug into theequation for a plane now...choosing t = 0 from L1 yields the point(3, 3, 8) so the equation of the plane is.-5(x-3)+1(y-3)+3(z-8)=0-5x+y+3z-12=0

Determine if there is a plane in mathbb(R)^3 which contains both lines l_1 and l_2 where l_1 : x= 2t + 3, y= t +3, z = 3t +8 and l_2: x=3s - 2, y = 3s - 1, and z = 4s+1.

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