Vectors and the Geometry of Space

Here are some general steps:1) Write the equations of line 1 and line 2 in vector form as R1(t) and R2(s).2) Set R3(u) equal to the given point A, which has coordinates (xo, yo, zo).3) Equate the corresponding components of R3(u) and A to get three equations in three unknowns (t, s, and u).4) Solve for t, s, and u using the three equations. This will give you the specific values of t and s that make it possible for R3 to go through A.5) Substitute those values into the equation for R3 to get the specific equation for the line that goes through A
  • #1
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Line 1 and line 2 are given by equation 1 and 2. Point A has coordinates (xo, yo, zo). Find the equation of line 3 which goes through A and crosses L1 and L2.
 
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  • #2
tmlfan_17 said:
Line 1 and line 2 are given by equation 1 and 2. Point A has coordinates (xo, yo, zo). Find the equation of line 3 which goes through A and crosses L1 and L2.
You failed to include equations 1 and 2.
 
  • #3
Equation 1 and 2 are not supposed to be given.
 
  • #4
tmlfan_17 said:
Line 1 and line 2 are given by equation 1 and 2. Point A has coordinates (xo, yo, zo). Find the equation of line 3 which goes through A and crosses L1 and L2.

Suppose line 1 is R1(t) = P + tD1
Line 2 is R2(s) = Q + sD2

where the P and Q are given points and the D's are given direction vectors.

Then a line from one to the other would be
R3(u) = R1(t) + u(R2(s)-R1(t))

Set that equal to your given point. You have three given coordinates of the point and three parameters s,t, and u to work with.
 
  • #5
I don't understand why you put R1(t) as the r0 vector in the R3(u) vector equation instead of just inputting the given point A there.
 
  • #6
tmlfan_17 said:
I don't understand why you put R1(t) as the r0 vector in the R3(u) vector equation instead of just inputting the given point A there.

The line R3 goes from the line with parameter t to the other with parameter s with t and s unknown. You have to choose s and t so that it's possible for R3 to go through A for some value of u. For most choices of s and t the line won't pass through A for any value of u.

[Edit -- Added] The general formula you get is long and messy but doing it for specific lines and points isn't too difficult.
 
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