>> Suppose we want to find parametric equations for a line that's parallel to a specified line but through a point? Find parametric equations [inaudible] a line through 3, 1, 3 and parallel to the line with equations X equal 4 minus 5 T, Y equals 7 plus 2 T. Z equal 2 plus 4 T. [Inaudible] when we look at this going backwards, we realize right way that this is equivalently in vector form. XYZ equals 4, 7, 2 plus T times negative 5, 2, 4. In other words here's our vector parallel to the line. And so if this vector is parallel to the line given here, and the line we want is parallel to this line, then this vector is also parallel to the line we want. It's parallel to our line, so we take V equal to negative 5 , 2, 4 and we use the point given. So that would be X, Y, Z equals 3, 1, 3 plus T times negative five, 2, 4. Or separating out the components in component form, rather in parametric form as a listed equation. X equals three plus negative five T, so that's three minus five T. ,Y equals one plus two T. , Z equals three plus four T. Now, you might notice that if I compare this system of parametric equations down here to the one up here, notice the terms with the T's are exactly the same, negative five T, negative five T, two T, two T, four T, four T. The only thing that changed are these constants. And so in our original equation we have 4, 7, 2. So in effect this line we're given pass through the point 4, 7, 2. And all we have to do is change the four, seven, two into three, one three, and that gets the answer.