Finding intersection of lines (vectors)

[TsAmE]

Find the intersection of the lines:

\(\displaystyle (7 + 2\lambda,3,-17 -7\lambda):\lambda \epsilon \mathbb{R})\) and

\(\displaystyle (-1 - \lambda, -5 - 4\lambda,10+3\lambda):\lambda \epsilon \mathbb{R})\)

Attempt:

1. \(\displaystyle 7+2\lambda = -1 -\lambda\)

2. \(\displaystyle 3 = -5 -4\lambda\)

3. \(\displaystyle -17 - 7\lambda = 10 + 3\lambda\)

For equation 3 I got

\(\displaystyle 8 =- 4\lambda\)

\(\displaystyle \lambda = -2\)

Now when I tried to sub in 2 for lambda in equation 1 to see if the LHS = RHS it doesnt:

\(\displaystyle 7 + 2(-2) = -1 - (-2)\)
\(\displaystyle 3 = 1\)

Why is this?

 

[galactus]

That means the lines do not intersect.

Setting the vectors equal and simplifying we get:

\(\displaystyle 8+3t=0\)

\(\displaystyle 8+4t=0\)

\(\displaystyle -27-10t=0\)

See the first two equations?. That gives it away.