Question

Determine whether the lines given by the vector equations r1=2i + 2j + 3k + s(i + 3j + k) and r2=2i + 3j + 4k + t(i + 4j + 2k) intersect. If they intersect, give the coordinates of their point of intersection.

Answer 1

The answer is I-j+2k at s=t=-1

Answer 2

i -j +2k . . . at s=t=-1

Explanation:

For the lines to intersect, there must be values of s and t that make the coordinates of r1 equal to those of r2.

Equating i coefficients, we have ...

2 +s = 2 +t

Equating j coefficients, we have ...

2 +3s = 3 +4t

Equating k coefficients, we have ...

3 +s = 4 +2t

The first equation tells us s = t. Using t = s in each of the other two equations, they become ...

2 +3s = 3 +4s ⇒ s = -1

3 +s = 4 +2s ⇒ s = -1

Then the point of intersection is where s = t = -1. That point is ...

(2 +s)i +(2 +3s)j +(3 +s)k = (2 -1)i +(2 -3)j +(3 -1)k

= i -j +2k . . . . the point of intersection