InterviewSolution
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InterviewSolution
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Show that the lines `vec( r) =(hat(i) +hat(j) -hat(k)) +lambda (3hat(i) -hat(j)) " and " vec( r) =(4hat(i) -hat(k)) + mu (2hat(i) +3hat(k))` intersect . Find the point of the intersection. |
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Answer» Comparing the given equations with the standard equations `vec(r ) =vec(a)_(1) =lambda vec(b)_(1) " and " vec(r) =vec(a)_(2) +mu vec(b)_(2)` we get `vec(a)_(1) =(hat(i) +hat(j)-hat(k)) ,vec(b)_(1) =(3hat(i) -hat(j))` `vec(a)_(2) =(4hat(i) -hat(k)) " and " vec(b)_(2) =(2hat(i) +3hat(k))` `:. (vec(a)_(2) -vec(a)_(1)) =(4hat(i)-hat(k)) -(hat(i) +hat(j)-hat(k)) =(3hat(i) -hat(j))` And `(vec(b)_(1) xx vec(b)_(2)) =|{:(hat(i),,hat(j),,hat(k)),(3,,-1,,0),(2,,0,,3):}|=(-3-0)hat(i)-(9-0)hat(j) +(0+2) hat(k)` `=(-3hat(i) -9hat(j) +2hat(k))` `:. |vec(b)_(1)xx vec(b)_(2)| =sqrt((-3)^(2)+(-9)^(2) +2^(2))=sqrt(94)` `:. SD =|((vec(a)_(2)-vec(a)_(1)).(vec(b)_(1)xxvec(b)_(2)))/(|vec(b)_(1)xx vec(b)_(2)|)|` `=(|(3hat(i)-hat(j)).(-3hat(i)-9hat(j)+2hat(k))|)/(sqrt(94))` `=(|-9+9+0|)/(sqrt(94))=0` Thus the shorest distance between the given lines is 0 Hence the given lines intersect. Thus for some particular values of `lambda " and " mu` we have `(hat(i)+hat(j)-hat(k)) +lambda (3hat(i)-hat(j))=(4hat(i)-hat(k)) +mu (2hat(i) +2hat(k))` `rArr (1+3lambda)hat(i) +(1-lambda)hat(j) -hat(k) =(4+2mu)hat(i) +(3mu-1)hat(k)` , `rArr 1+3lambda =4 + 2 mu , 1 - lambda =0 " and " 3 mu -1 =-1` `rArr lambda=1 " and " mu=0` Thus the position vector of the point of intersection of the given lines is given by `vec(r ) =(hat(i) +hat(j) -hat(k)) +(3hat(i) -hat(j))` [putting `lambda=1`] ,i.e., `vec(r ) =(4hat(i)-hat(k))` Hence the point of intersection of the given lines is P(4,0,-1) |
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