Question
Poini} Give vector parametric equatian for the line through the point (-4,1,0) that Is parallel to the Iine (-2 ~ 5t,4 + 3+,5): Lt)
poini} Give vector parametric equatian for the line through the point (-4,1,0) that Is parallel to the Iine (-2 ~ 5t,4 + 3+,5): Lt)
Answers
Show that $\mathbf{r}_{1}(t)$ and $\mathbf{r}_{2}(t)$ define the same line, where
$$
\begin{array}{l}{\mathbf{r}_{1}(t)=\langle 3,-1,4\rangle+ t\langle 8,12,-6\rangle} \\ {\mathbf{r}_{2}(t)=\langle 11,11,-2\rangle+ t\langle 4,6,-3\rangle}\end{array}
$$
Hint: Show that $\mathbf{r}_{2}(t)$ passes through $(3,-1,4)$ and that the direction vectors for $\mathbf{r}_{1}(t)$ and $\mathbf{r}_{2}(t)$ are parallel.
This exercise were given the point p their own minus five three and the Vector V, and she's equal to 20 minus four. And we want to find the equations for a line that passes through P is parallel to be so to get started, X is equal to zero plus two t because the first coordinate zero in the first component of these two, which is just equal to T. Why illegal to minus five plus zero t or just minus five? Because zero will cancel out whatever team put there. And finally, Z is equal to three months for tea, and there is no simplification we can do for that. So these three equations describe a line that passes through the 30.0 75 3 and his parallel to the vector to their own minus four.
Hi. So in this exercise we have Uh part of the dramatic interpretation of linear algebra. So here we have a point to find in the in the coordinates to -1 and a picture that will determine the the direction of a line. So basically you can define a line by fixing some point in the skin is the point P. And then given a direction in this case is given by the vector B. Because this line is going to be parallel to this vector. So if we put uh this picture in the point B, we can define an infinite line and we need to find the premature representation of this life that's called L. So L will be given as a parametric parametric equation. A vector that depends on T equals to the point plus the parameter T. Times the direction. So that's the way to construct a line parametric lee. You fix a point and then you give it that direction. So in this case the direction is even by this parallel vector. The So let's summarize the data. So we have here 2 -1 Plus T, Times -4 minus two. And then after seven days we obtain 2- for tea -1 -2 T. And this is the equation of the line.
Hello there. So in this case this occasion we have a point and a spectrum. With these two data. We need to define the line that passes through this point P. That is parallel to this vector B. So technically when you have some pro vector and a point and you went to the final line that pass through that point. This problem vector will actually corresponds to the direction that we're going to get. So this vector B will define the direction where we're going to extend online from the point. So in that sense, the line in this case the parametric equation of the line is defined as the point P. Where is where we're going to start plus T. Times the vector of direction. Here she is a free parameter that takes values on the reels. And that's what these volunteers going to do is extending or shrinking this vector V. The direction. So we start to be and then we start to extend it or contracted. Things are even inverting the direction. So you can see that for every T. We're going to have a point that lies on this that is parallel to the vector, so that's more or less the geometric idea of this exercise, but it's really easy to solve. We just need to replace the data so we have the P is equal to minus 934 and the victor Plus three times the actual be. That give us the direction 160 And from this we obtain the parametric equation -9 -T. Three plus six times T four.
Hi. So in this occasion we have a simple exercise to determine the line that passes through this point and it's parallel to this vector. So everything in algebra has a geometric analog. So these exercise focus on the geometric part. So basically it means that we have here a point P. And the vector B will determine the direction that we're going to take it from this point. And then if this vector determine the direction then we we we we can define a line parametric equation for a line that passed through this point is parliament to this vector and extensive infinity. Okay, so that's the geometric interpretation of this exercise. And the way to construct this line we're going to call L is really easy. So L will be the term mind by vector X. Greeting us. The point where this line is passing through plus the parameter T. So this is a parameter equation, picnic grape properly will be the point X. That depends on this parameter T. Times the vector or the direction in this case is problem to this vector V. So we write in this way so this is equal To have in -41 plus Tea time, 09 is eight. And then we can solve technically this, so we obtain the vector, the barometric factor that pass through this point and goes in this direction or is proud to this venture. So the solution is minus 41 minus eight.