Question

Show that the lines (x - 1)/4 = (2 - y)/6 = (z - 4)/12 and (x - 3)/-2 = (y - 3)/3 = (5 - z)/6 are parallel.

Answer 1

We observe that the straight line (x - 1)/4 = (2 - y)/6 = (z - 4)/12 is parallel to the vector 4\(\hat {i}\) - 6\(\hat {j}\) + 12\(\hat {k}\) and the straight line (x- 3)/-2 = (y - 3)/3 = (5 - z)/6  is parallel to the vector.  -2\(\hat {i}\) + 3\(\hat {j}\) - 6\(\hat {k}\).

Since  4\(\hat {i}\) - 6\(\hat {j}\) + 12\(\hat {k}\), =  -2(-2\(\hat {i}\) + 3\(\hat {j}\) - 6\(\hat {k}\)) the two vectors are parallel, and hence the two straight lines are parallel.