Is the line through $ (-4, -6, 1) $ and $ (-2, 0, -3) $ parallel to the line through $ (10, 18, 4) $ and $ (5, 3, 14) $?
we are given two lines that were asked to become that they're parallel. So we're told the first line goes through the point. Negative four. Negative six one and the points negative, too. Zero negative three. And the second line goes through the points. 10, 18, 4 and 53 14. Well, check out. These two lines are parallel. Will want to find a vector on each line and then check. These vectors are parallel, of course. No. The two vectors are parallel if, and only if the ratio of their correspondent components is a constant. So we know two points in the first line, so vector joining them on the first line. They'll call V one. Yeah. Has components negative two minus negative. Four zero minus negative. Six and negative three minus one. This is positive, too. Six. Negative four. The second line contains the other two points and so contains the Vector V two, with components five minus 10, three, minus 18 and 14 minus four. This simplifies to negative five negative 15 and 10. Now let's find the ratio of the corresponding components well. The ratio of the X terms is to over negative five the ratio of the wide components is six over negative 15 and the ratio of the Z components is negative. 4/10. Notice that these are all equal to negative 2/5. Since this is constant, it follows that the two lines L one and L two are parallel.