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The Vector Equation of a Line - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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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