Parametric equations and lines

In summary, in order to determine if any of the given lines are parallel or identical, we can compare their respective vector equations and see if they are scalar multiples of each other. If they are, then they are either parallel or identical. To determine if they are identical, we can choose a point that satisfies one equation and see if it also satisfies the other equations. If it does, then the lines are identical. If not, they are still parallel, but not identical.
  • #1
Brigada
2
0

Homework Statement


Determine if any of the lines are parallel or identical
L1 (x-8)/4 = (y+5)/-2 = (z+9)/3
L2 (x+7)/2 = (y-4)/1 = (z+6)/5
L3 (x+4)/-8 = (y-1)/4 = (z+18)/-6
L4 (x-2)/-2 = (y+3)/1 = (z-4)/1.5

Homework Equations


L1 pt(8,-5,-9) V<4,-2,3>
L2 pt(-7,4,-6) V<2,1,5>
L3 pt(-4,1,-18) V<-8,4,-6>
L4 pt(2,-3,4) V<-2,1,1.5>

The Attempt at a Solution


I know that if the vectors are scalar multiples, they are either parallel or identical. What I don't know, is after I find out that V(L3) = -2*V(L1). How do I determine if they are parallel or identical. I assume that since one vectors k value is 1.5, it is some multiple of another line, if not they wouldn't have given a 1.X. So L1 L3 L4 are parallel but how could I find if they were identical or not?
 
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  • #2
After careful consideration, and pulling my head out of the book to think logically, I realized after I find the parallel lines, if a point I chose that satisfies one equation, also satisfies another line equation, they are identical, if its not on the line, it's still parallel, just not identical. That was 1.5 hours wasted on a brain fart!
 

Related to Parametric equations and lines

1. What are parametric equations and how are they different from regular equations?

Parametric equations are mathematical expressions that describe the relationship between two or more variables. Unlike regular equations, which typically only involve one independent variable, parametric equations involve multiple independent variables and can represent more complex curves and shapes.

2. How are parametric equations used in real-world applications?

Parametric equations are used in a variety of fields, such as physics, engineering, and computer graphics, to model and analyze complex systems and phenomena. For example, they can be used to describe the trajectory of a projectile, the motion of a pendulum, or the shape of a curve in a 3D animation.

3. What is the difference between a parametric equation and a parametric line?

A parametric equation can represent any type of curve or shape, while a parametric line specifically describes a line in a two-dimensional or three-dimensional space. A parametric line is typically written in the form x = a + bt and y = c + dt, where a, b, c, and d are constants and t is the parameter.

4. Can parametric equations be solved for specific values?

Yes, parametric equations can be solved for specific values by plugging in the desired value for the parameter. For example, if the parametric equation for a curve is x = 2t and y = 3t², plugging in t = 2 will give the coordinates (4, 12) on the curve.

5. What is the relationship between parametric equations and polar coordinates?

Parametric equations and polar coordinates are both ways of representing points and curves in a two-dimensional space. While parametric equations use two independent variables (usually denoted as x and y), polar coordinates use a distance from the origin (r) and an angle (θ) to describe a point. Parametric equations can also be converted to polar coordinates and vice versa.

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