Finding equations of two lines

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In summary, the task is to find two lines in R3 that are not parallel and do not intersect, using the given equations and attempted solution. The attempted solution of (1,1,1) and (2,3,4) does not satisfy the conditions as they are not proper line vectors and do not have a starting point. The solution requires specifying a point on each line and using a parameter such as t to construct the lines. The final step is to find skew lines that do not intersect.
  • #1
mrroboto
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Homework Statement



Find 2 lines in R3 that are not parallel and do not intersect.

Homework Equations



orthogonal: a.b=0
parallel: a=tb

The Attempt at a Solution



(1,1,1) and (2,3,4)

(1,1,1).(2,3,4) = 7
and there does not exist a t such that
t(1,1,1) = (2,3,4)
 
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  • #2
Is that all you're given? Any 2 lines would do?

Your answer doesn't include the starting point of the line vector it's supposed to be in: (x0,y0,z0) + t(a,b,c). The point (x0,y0,z0) isn't specified. That's why you haven't shown why they do not intersect.
 
  • #3
If fact, your answers are not lines at all! What you give look like points but I guess you mean them as vectors- and I presume you mean them as "direction" vectors for the lines. The fact that "(1,1,1).(2,3,4) = 7" shows that they are not perpendicular (not really relevant) and the fact that "there does not exist a t such that t(1,1,1) = (2,3,4)" shows that they are not parallel. Neither of those means they do not intersect.
As defennnder said, you need to write them as lines with a parameter such as t. To do that, you will have to specify a point on them- and whether or not they intersect will depend on that point. For example, the lines [itex]\vec{r_1}= t\vec{i}+ t\vec{j}+ t\vec{k}[/itex] and [itex]\vec{r_2}= 2t\vec{i}+ 3t\vec{j}+ 4t\vec{k}[/itex] have direction vectors [itex]\vec{i}+ \vec{j}+ \vec{k}[/itex] and [itex]2\vec{i}+ 3\vec{j}+ 4\vec{k}[/itex] respectively and intersect at (0, 0, 0). You need to find skew lines.
 
  • #4
First try to get the geometrical picture clear. Can you describe in words two lines that would do?
 

Related to Finding equations of two lines

1. What is the formula for finding the equation of a line?

The general formula for finding the equation of a line is y = mx + b, where m is the slope of the line and b is the y-intercept.

2. How do you find the slope of a line?

The slope of a line can be found by calculating the change in y-coordinates (rise) over the change in x-coordinates (run) between any two points on the line. This can be expressed as (y2 - y1) / (x2 - x1).

3. What is the point-slope form of the equation of a line?

The point-slope form of the equation of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. This form is useful for finding the equation of a line when given a point and the slope.

4. Can two lines have the same equation?

No, two lines cannot have the same equation. Each line has a unique equation that describes its slope and y-intercept. However, two lines can have equations that are equivalent or parallel, meaning they have the same slope but different y-intercepts.

5. How do I find the intersection of two lines?

The intersection of two lines can be found by setting the two equations equal to each other and solving for the values of x and y that satisfy both equations. The resulting point of intersection will be the coordinates where the two lines intersect.

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