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Find vector equation and parametrc equations for the line: (Use the parameter [. ) The line through the polnt (0, L5, 12) and parallel to the line 4t,Y = 6 36,2 = 3...

Question

Find vector equation and parametrc equations for the line: (Use the parameter [. ) The line through the polnt (0, L5, 12) and parallel to the line 4t,Y = 6 36,2 = 3r(e)Xt) , z(t))

Find vector equation and parametrc equations for the line: (Use the parameter [. ) The line through the polnt (0, L5, 12) and parallel to the line 4t,Y = 6 36,2 = 3 r(e) Xt) , z(t))



Answers

Find a vector equation and parametric equations for the line.

The line through the point $(6,-5,2)$ and parallel to the vector $\left\langle 1,3,-\frac{2}{3}\right\rangle$

All right, we've got a question here. We want to find the vector equation and Parametric equations for the line. So basically the way that we do this as we write out our standard form, which is the same thing as our equals R not was t. V. Alright, here we write. Our are not as our original vector. Um, excuse me as our as the point that it passes through. We know that the point it passes through six. Negative five and two is We read that ad, but you six I'm on negative five two, and we're gonna add t multiplied by the V here, and we'll be here is the, uh is the vector that we're told it's parallel to. So we're told that it's parallel to the victor. 13 negative. Two thirds. All right, that's what we do now is we start to combine our eyes. Jay's in our case here. We know our eyes are going to be the first, um uh, number represents represents the I. The second ever represents the J and the 3rd 1% k values. So if you write this out, we'll write it as six plus a one times t here and those of the r I values make it black for cheap You got a plus sign, right? And then we'll have that as our I values. Then we're gonna add our negative five plus our three times t and that'll be our J values. And finally, the K values would be too minus, uh, two thirds times t okay. And those are the representative of our K values. Alright, Now we write out our parametric equations are parametric equations are going to be the equations that represent R I, J and K are X represents our values. We would say that our X equation is just six plus t right, And our wider question is what's going to represent our J values? So we just take what's in our parentheses there and we say, Well, why is he puts a negative bye? What's three? Gee, And then finally, for our k values, we write that as an equation with equal to see who says he is equal to extend the brackets here, which is to minus two thirds, Chief. All right, and that's gonna be our final answer is there? And our final answer for our our equation of here. All right, well, I hope that clarifies the question there. And thank you so much for watching.

No I got a problem for again. The problem for is the same thing. It passes to the going to Judo 14 and -10. And again it's parlor too. X equals 2 -1 plus two. T Why equal to 6- Treaty? And G equals 23 plus 90. Right now again with the same thing for the parameter equation. What we do can you do the same thing? This is my remember and this the vicar be here is two -3 and nine so He can parliamentary question. It just takes -0/2. The same as Y -14/-3. The same as G plus 10 overnight. And again to use a secret to T. So access to T. Why is -3 t plus 14 And G. is 90 -10. This is how the question looks like. And for the better equation like we said before this is a not some T. V. So this is due to wake up plus 14 Jacob Afghanistan K cap nasty times to wake up minus T. Jacob plus nine. Kick and dissect. That's the answer.

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.

On the line so that contents upon extra y 00 and it has the dash with the vehicle to A, B and C, and then we have the better a question on the farm on they go to the X zero y 00 plus the three times the A, B and C here and in discussion were given the extra wiser Thistle ICO Thio, too. 2.43 point five and a better direction equal to the three to minus one three to minus one. Therefore, the vector equation Wilken Fungal B R t. It will equal to the two coupon far 2.5 and then plus with the tee times with the three to minus one, we can turn this one into the environment. E question. On the form X Y Z equal to the two plus three T and Cuban four plus duty 3.5 minus day


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