Are these vectors the same?

Question

$r = 2j -k + \lambda(i +2j -3k)$

$r'= 5i +6j -10k +\mu(2i +4j -6k)$

If so, could you show me how they are equal

$i: 5 = \lambda $

$j: 6 = 2 + 2\lambda $

$k: -10 = -1 -3\lambda $

Seems like lambda's value are different so for me it doesn't look like they are the same line but the answer says it is.

Answer 1

The first vector is $$<\lambda,2+2\lambda,-1-3\lambda>$$

The second one is $$<5+2\mu ,6+4\mu,-10-6\mu>$$

Solving for $\lambda$ and $\mu$ results in an inconsistent system.

There are no values of $\lambda$ and $\mu$ to make them the same vectors.

Thus my answer is no they are not the same vectors.

Answer 2

If so, then we have $$\vec{r}=\vec{i}\lambda+\vec{j}(2\lambda+2)+\vec{k}(-1-3\lambda)$$ $$\vec{r'}=\vec{i}(5+2\mu)+\vec{j}(6+4\mu)+\vec{k}(-10-6\mu)$$ so it would be $$10+4\mu+2=6+4\mu$$ so$12=6$ they are not the same.