Question 1121153
a.(i)  You are correct.   The formula, given points P(x1,y1,z1) and Q (x2,y2,z2) is   v = < x2-x1, y2-y1, z2-z1 >. for a vector that points in the direction from P toward Q.    <br>

a. (ii)  They want a parametric representation of the line.  What you wrote is essentially correct, however, be aware that you are specifying vectors, not just coordinates.   Sure, once you evaluate a vector equation to a resultant < a,b,c > you can say the tip of the resultant vector is at the point (a,b,c), but be sure to keep vectors straight.     Here you can take one point as the start {{{ r[0] }}}  (say P) and then use the vector you found in (a (i)) multiplied by a parameter t:   r = < -1, 6, -1 > + t* < 1,-2,2 >,  so  {{{ highlight( matrix(1,10, "r  ", " = ", "  <",  -1 + t, ",",  6 - 2t,",",  -1 + 2t, " >", "")  )  }}}  ( here, -1+t is the x component,  6-2t is the y component, and -1+2t is the z component ).   t varies from {{{-infinity}}} to {{{infinity}}} <br>

b.  The 2nd line has form:   {{{  r[2] }}} = <4+3s,  2, -1-4s>

The "dot" product of two vectors is   A &#8901; B  =  {{{ A[x]B[x] + A[y]B[y] + A[z]B[z]  = abs(A)abs(B) cos((theta)) }}}   where {{{theta}}} is the angle between the two vectors,  {{{ abs(A) = sqrt(A[x]^2 + A[y]^2 + A[z]^2) }}} and  {{{ abs(B) = sqrt(B[x]^2 + B[y]^2 + B[z]^2) }}}, gives a SCALAR measure of how much vectors A and B point in the same direction.     So you have all the numbers needed.   Let A be the vector PQ you found, and let B be the vector <3,0,-4> (the part that 's' scales).    This is so because  s<3,0,-4> and t<1,-2,2>  have the same angle between them as the two lines PQ and L2.  
 (  If this last fact is not clear, watch this video  https://www.youtube.com/watch?v=PyPp4QvQY3Q   ) 

You will compute the numbers and solve for {{{ cos(theta) }}} <br>

c.  At the point of intersection,  {{{ r[x] = r[2x] }}}, {{{r[y] = r[2y] }}} and {{{ r[z] = r[2z] }}}  (where {{{r[2x]}}} is supposed to be r sub 2, sub x,  etc.).   Each of the lines (a(ii)) and (b)  can be thought of as the tip of a vector traced out as t (s) varies.   <br>

Setting x components equal:   -1+t = 4+3s  
Setting y components equal:   6-2t = 2  
Setting z components equal:  -1+2t = -1-4s <br>

The y component equation gives t=2, substituting into the x component equation
gives  s=-1  (and these values work in the z component equation, so it is a good solution).
To find the coordinates, substitute t=2 in part a(ii)  and it should match the coordinates when
you set s=-1  in part b.  You should find they intersect at (1,2,3).<br>

Hopefully Theo will agree with these solutions.  Maybe he can explain it better or make any needed corrections.  I was rusty on it as well.