Vectors and the Geometry of Space

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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
mathman
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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.
You failed to include equations 1 and 2.
 
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Equation 1 and 2 are not supposed to be given.
 
  • #4
LCKurtz
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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.
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.
 
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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
LCKurtz
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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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