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.