Vectors and the Geometry of Space

[tmlfan_17]

Line 1 and line 2 are given by equation 1 and 2. Point A has coordinates (x_o, y_o, z_o). Find the equation of line 3 which goes through A and crosses L1 and L2.

 

[mathman]

Line 1 and line 2 are given by equation 1 and 2. Point A has coordinates (x_o, y_o, z_o). Find the equation of line 3 which goes through A and crosses L1 and L2.

You failed to include equations 1 and 2.

 

[tmlfan_17]

Equation 1 and 2 are not supposed to be given.

 

[LCKurtz]

Line 1 and line 2 are given by equation 1 and 2. Point A has coordinates (x_o, y_o, z_o). 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.

 

[tmlfan_17]

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.

 

[LCKurtz]

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.