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Find a vector equation and parametric equations for the line.

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

The vector equation:

$\mathbf{r}=<6+t,-5+3 t, 2-\frac{2}{3} t>$

The parametric equation:

$x=6+t \quad y=3 t-5 \quad z=2-\frac{2}{3} t$

Vectors

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Johns Hopkins University

Campbell University

University of Nottingham

Idaho State University

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.

The University of Texas at Arlington

Vectors