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The Vector Equation of a Line - Concept
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We can use vectors to create the vector equation of a line. In order to create the vector equation of a line we use the position vector of a point on the line and the direction vector of the line. In order to find the direction vector we need to understand addition and scalar multiplication of vectors, and the **vector equation of a line** can be used with the concept of parametric equations.

Now we do know how to come up with the equation of a line using a lot of different methods for example you could be given a point and a slope and use the point slope formula to come up with the equation of a line. With the vector equation line is very similar take a look at our example here, we start with a point just and arbitrary point x0, y0 and then we define its position vector. Remember a position vector, is a vector going from the origin to a given point and we'll call that our 0. Next let's define a direction vector v this will give us the direction that our line will go and notice I've drawn v so that it's head to tail with r so that I can add the 2 together and I get r 0 plus v.

Now if I do this again, if I add a second v I get r0 plus 2v so this is still r0 plus v this is r0 plus 2v and I can do it again. I can add another v and get r0 plus 3v or I can take r0 and add a negative v to it. But you'll notice that all the points that I get that correspond to these position vectors are all on a line. And so the idea here is I can get any point on this line by adding an appropriate scalar multiple of v to r so r plus some scalar multiple of v is going to give me any point on this line and that suggests this equation. So this will be the vector equation for line r equals r sub 0 plus t times v. So the givens what you need to have to get the vector equation of a line is r0 some position vector for a given point and v a direction vector that tells you the direction that the line goes in.

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