in this video we're going to talk about how to find the vector equation parametric equations and symmetric equations of a line if we're given the point that's on the line and the vector that's parallel to the line and in another problem we're going to talk about how to do the same thing if we're given two points on the line so let's go over the basics behind this so let's say we have a 3d coordinate system where this is x y and z and we're going to have a generic line now a line is defined by two things we need a point on a line and the direction of the line if we have those two things we could define any line so let's say that we have this point which is it has the values x zero y zero or z naught this is going to be the position vector r naught and let's extend this point by another vector let's call this vector g if we add those two vectors if we take the vector sum of those two vectors it will give us the position vector r which will define the vector equation of the line so r is the vector sum of these two vectors now vector g is going to be parallel to vector v because they're both parallel to each other g is going to be some scalar quantity multiplied by v so we're going to say it's t times v so thus we have the vector equation of a line which is r naught what just happened there it's r naught plus t times v so this is the formula that we're going to use in order to write the vector equation for this problem now some other things that you want to keep in mind is that the position vector r naught is equal to x naught y naught and z naught and v is defined by these values so in this problem a is three b is negative two c is five x naught is two y naught is four z naught is negative three so let's go ahead and work on this problem so r naught is based on these values so it's going to be 2i plus 4j minus 3k and then plus t times vector v which we can replace it with that vector v is going to be parallel to the line so that's 3i minus 2j plus 5k now i'm going to distribute t and then i'm going to pair up all of the the i's the j's and the case together so i'm going to have 2i plus 3i t and then let's put this over here 4j minus 2jt and then minus 3k plus 5kt now the next thing i'm going to do is factor out i j and k so i'm going to have 2 plus 3t times i and then plus this is going to be four minus two t times j and then plus negative three plus five t times k so this right here is the vector equation of the line so that's the answer now we need to move on to the next part we need to find the parametric equations that correspond to this line so this is x this is y and here we have a z so to write the parametric equations it's going to be x is equal to 2 plus 3t and we're going to have y is equal to 4 minus 2t and z is going to be equal to negative three plus five t now the generic formulas for the parametric equation are as follows it's x is equal to x naught plus a t y is equal to y naught plus b t and z is equal to z naught plus c t so all you need in order to write the parametric equations is the point x naught y naught z naught which is what we have here and then the vector v which is uh a common b comma c which we have that here so these are the parametric equations which is the answer for part b now for part c we need to find the symmetric equations to do that we need to eliminate the parameter by solving for t for each of these equations and then we can set them equal to each other so what we're going to have is x minus x naught divided by a and that's going to be equal to y minus y naught divided by b which is equal to z and minus c naught divided by c so that's the formula that will give us the symmetric equations x naught is two a is three y naught is four b is negative two z naught is negative three so z minus negative three that's gonna be z plus three and c is five so these are the symmetric equations that correspond to the line that we're dealing with so that's it for number one number two find the vector equation parametric equations and symmetric equations for the line that passes through the points one three negative two and four one five now in order to define a line we need a point and a vector or which will define the direction of the line we don't have a vector but we could find a vector using the two points that we have so let's call this point a and point b so our vector will be vector a b and that's going to be the difference between points b and a so 4 minus 1 that's going to be three one minus three is negative two and five minus negative two that's gonna be five plus two which is seven so that's our vector now point a we're gonna say that is our standard point so thus we have the values x naught y naught and z naught and this is going to be a b and c so at this point we have everything that we need in order to find the vector equation the parametric equations and the symmetric equations now go ahead and try this problem based on the last example so let's start with the vector equation using this formula so r naught is going to be based on the point one three negative two so as a vector it's going to be 1 i plus 3 j minus 2 z and then it's going to be plus t times vector v which is 3i minus 2j plus 7k so now let's factor out i which is going to be based on those two quantities so it's going to be 1 plus 3t times i and then factoring out j it's going to be based on those values so it's plus three minus two t times j and then we're going to take out k for some reason i put z this should be a k here so that was a mistake so taking out k from those values it's going to be negative 2 plus 70 times k so that's going to be the vector equation now let's move on to part b let's write the par excuse me the parametric equations but we're going to use the formula to get the answer even though we could find it directly from the vector equation so x naught is one a is three so we have this x is equal to one plus three t y naught is negative two now i take that back and y naught is three b is negative two so we have that and then z naught is negative two c is seven so these are the parametric equations for the line now let's write the symmetric equations that describe the line so using the formula it's going to be x minus x naught over a which equals y minus y naught over b which equals z minus z naught over c so x naught is one a is three y naught is three b is negative two z naught is negative two so z minus negative two that's z plus two and c is seven so these are the symmetric equations so that's it for this video so now you know how to find the vector equation the parametric equations and the symmetric equations for a line in three-dimensional space