Is finding the intersection of two lines in R4 using the same method as in R3?

In summary, to find the intersection of two lines in R3 or R4, you would set the equations for the two lines equal and solve for the unknown values. The typical behavior in R3 is for lines to be skew, while in R4 it is even less likely for two lines to intersect.
  • #1
theRukus
49
0
To find the intersection of two lines in R3, you set the lines equal, right?
[a,b,c] + d[e,f,g] = [h,i,j] + k[l,m,n]
Then split these into three equations,
1. a + d(e) = h + k(l)
2. b + d(f) = i + k(m)
3. c + d(g) = j + k(n)

And solve for k and d, correct?

If k and d are consistent, then these values are used to find the point of intersection.

My question is: Is this same method used in R4, with two lines of 4 dimensions?
 
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  • #2
Probably you could get some help after explaining your notation...
 
  • #3
Yes, it is exactly the same. You would have one line given by, say,
[tex]\begin{pmatrix}x_1 \\ x_2 \\ x_3 \\ x_4\end{pmatrix}= \begin{pmatrix}a_1t+ b_1 \\ a_2t+ b_2 \\ a_3t+ b_3 \\ a_4t+ b_4\end{pmatrix}[/tex]
and the other by
[tex]\begin{pmatrix}x_1 \\ x_2 \\ x_3 \\ x_4\end{pmatrix}= \begin{pmatrix}c_1s+ d_1 \\ c_2s+ d_2 \\ c_3s+ d_3 \\ c_4s+ d_4\end{pmatrix}[/tex]
then you would set them equal getting 4 equations in the two unknown values s and t:
[itex]a_1t+ b_1= c_1s+ d_1[/itex]
[itex]a_2t+ b_2= c_2s+ d_2[/itex]
[itex]a_3t+ b_3= c_3s+ d_3[/itex]
[itex]a_4t+ b_4= c_4s+ d_4[/itex]

Of course, it would be unlikely for 2 numbers to satisfy all 4 equations!

In the plane, the "typical" behavior is for lines to intersect- the only non-intersecting lines are parallel lines, a very unusual situation. And, in two dimensions, we are solving two equations in two unknowns- there would not be a unique solution only if the determinant of the coefficient matrix were 0.

In three dimensions, the "typical" behavior is for lines to be skew- not intersecting. In order to be able to solve three equations for two unknown values, the equations must NOT be independent.

And the situation is worse in four dimensions. Two lines intersecting is a very special situation indeed.
 

FAQ: Is finding the intersection of two lines in R4 using the same method as in R3?

What is the intersection of lines in R4?

The intersection of lines in R4 refers to the point where two or more lines in four-dimensional space intersect. It is the common point shared by all the lines.

How is the intersection of lines in R4 calculated?

The intersection of lines in R4 can be calculated by setting up a system of equations that represent the lines and solving for the variables. This can be done using various methods such as substitution, elimination, or matrices.

Can the intersection of lines in R4 be empty?

Yes, the intersection of lines in R4 can be empty if the lines are parallel or do not lie in the same four-dimensional space. In such cases, the lines do not have a common point of intersection.

What is the significance of the intersection of lines in R4?

The intersection of lines in R4 is significant in solving systems of linear equations and finding solutions to real-world problems. It also helps in understanding the relationship between different lines in four-dimensional space.

Are there any real-life applications of the intersection of lines in R4?

Yes, the intersection of lines in R4 has various real-life applications such as in computer graphics, engineering projects, and navigation systems. It is also used in solving optimization problems and analyzing data in fields like economics and physics.

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