Find both the vector equation and - the parametric equations of the line through (0,0,0) that is parallel to the line r = (1 6t,3 46,7where ( = corresponds to the g...

Equations of lines Find both the parametric and the vector equations of the following lines. The line through (0,0,1) in the direction of the vector $\mathbf{v}=(4,7,0)$

This exercise for the point P 3 to 1 on the Vector V, which is equal to zero, longs for to. We want to find a line that passes through P and eyes panel TV and want to find the equations for that line. So we have, as we saw before, X is equal to 30 T or just three. Why is equal to two minus 40? There's no simplification to be done. For that Z is equal to one plus two t where t is the parameter for the equations. So these are the car metric equations that describe the line that we wanted.

Hi in this occasion we have a point and a vector. So we need to find the the line that passed through this point and this parallel to the vector B. So let's say that here we have the vector B. And we have a line that passed through this point B. That is parallel to that effect. Let's call this line out. But basically the way to construct this line is too kind of translating this vector to this point to obtain the line and that is technically taking a point and then given a direction. So basically this problem vector here, what's going to do is give us a direction. So in that sense, the line, the diplomatic equation of the line is the finest fault. We fix the point plus two times and the direction. Why? Because this perimeter here will delegate the vector V infinitely in one direction and in the other direction. That's why this city takes any value in the reels. So with this information, this is a parametric equation of this line, so we just need to change the data. But here not something this point P corresponds to the origin. So the parametric equation at the end becomes just taking T. Times the vector B. And that corresponds to t minus 301 And in case I don't want to have like an equation, you can just multiply it'd by each of the components in the better, so minus three T zero anti. And this is a parametric equation of this line.

This problem was given a point at a direction vector. They were asked to write a parametric equation um of the line that goes to that point in that direction. And then the symmetric equations that also go from that line. Okay so this is to get the parametric equation, it's simple. We just take the point and then at the direction vector times, time or some parameter here. So you can see that the certainly passes through this point because when T equals zero well at this point and then as we move along t we go in this direction. So then the symmetric equations basically just break this out into scaler components and saw for teeth in each case. So um T equals in the X. Um coordinate you get T equals x minus 4/3. And the y coordinate T. Cause why minus 5/2. And then the z coordinate equals T. Equals z minus six, all over one. So these are all equal to t. Then they all must be equal to one another. These are parametric equations. Now these aren't necessarily unique. We could um you know multiply them all by some value, divide them all by some value, add some value to all of them whatever. But there are mathematically equivalent. So in the second case again we'll give her these two points and you can quickly write down what the parametric equation for the line that goes through this point into this direction. And then the symmetric equations we can solve the X. One. We have T equals X plus one all over minus two. The Y one notice we have a slope that is zero. So we don't have that. We can't solve fatigued. He doesn't occur in that equation but we just have white those three. And then there's the equation we have Z plus six, all over five. So these are the parametric equations in that case. In this case we have 2.111 in the direction of minus 10 minus 100 minus 1000. And so we have the permission form of the line goes to this 0.111 And I pull the minus sign out. So the direction is just 110. 100. 1000 and then multiplied by T. Or you could actually put a plus here and then we change the silent E. So again, none of these are unique because we could always re parameter rise with the change of variables. Then our parametric person metric equation kinds of being x minus one. All over minus 10 must equal y minus one. All over minus 100 and z minus one equals z minus one. All over minus 1000. And of course we could put a minus sign in front. Yeah. If we want to make things a little bit nicer. So that then we have these equations here and finally we have the point minus two to minus two. So we have our offset for our parametric online. And then we have the correction vector, which is just the slope in each case. And so then we can solve for T. So T equals X plus two. All over seven. And also because why minus two? All over minus six. And it also equals um that should be is the X Z plus two all over three. So if those are all equal to T, then they all must be equal to one another. So we have these are the symmetric equations for the for that government that prime motorized or that describe the line describe this line here, a line that goes at this point and is in this direction.

For the point p 000 on the Vector V, which is equal to negative for 35 Let's find our pirate trick equations that describe a line going through P and being parable to V. Oh, so to get started, X is equal to zero plus negative for team or just negative for it's team Lying is equal to zero close three T or just three T and similarly Z is equal to zero plus five t, which is just five t as you can see because P is the origin, there is no translation off the line. We're just taking all factors, all points that are parallel to V or other that can be obtained by scaling B and looking at the end point. So these are the parametric equations that describe his life.