Dear maggiec,
Take, $\displaystyle L_1:\underline{r_{1}}~and~L_2:\underline{r_{2}}$
First write the vector equation of AB. ($\displaystyle \overline{AB}=\underline{r_{2}}-\underline{r_{1}}$)
Since AB is perpendicular to L1 and L2, $\displaystyle \overline{AB}.\underline{r_{2}}=0~and~\overline{AB }.\underline{r_{1}}=0$
By the resulting equations you will be able to find the corresponding $\displaystyle \lambda~and~\mu$ values.
Hope you would be able to continue.
Proceeding with the method you have described is taking a very long time and leaving me $\displaystyle \lambda~and~\mu$ to the power of 4 that are very hard to work out! Especially as this question is form a non-calculator exam.
I think there must be and easier way to work this out.
The way I had done it, is by working out $\displaystyle \overline{AB}=\underline{r_{2}}-\underline{r_{1}}$
and then establishing a relationship between the line $\displaystyle \overline{AB}$ and the position vector (2,1,2)
This is possible since before i had proved that this position vector was perpendicular to both $\displaystyle L_1:\underline{r_{1}}~and~L_2:\underline{r_{2}}$ and hence it must be parallel to the line $\displaystyle \overline{AB}$
with this I equated the i and k components of $\displaystyle \overline{AB}$
and doubled the component j of $\displaystyle \overline{AB}$
However the numbers I'm getting seem really odd.
Dear maggiec,
For the moment I can't think of an easy method. But if you follow my method you will get $\displaystyle \lambda^2~and~\mu^2$ terms not $\displaystyle \lambda^4~and~\mu^4$ terms.
The method you have used is incorrect. It is not reasonable to equate i and j components of AB and the vector (2,1,2)
Unfortunately that isn't working. I have looked at the product and $\displaystyle \lambda~and~\mu$ both become 0.
Do you want to check that my $\displaystyle \overline{AB}$ vector is correct?
i get: $\displaystyle (13.5+7mu-3lambda)i + (-8.5+8mu+4lambda)j + (8.5-11mu+lambda)k$
Hello maggiecI agree with your working, and also your method - which is to say that this vector is parallel to $\displaystyle 2\vec i + \vec j + 2\vec k$.
I have done this, and I also get rather nasty numbers. I get the following equations for $\displaystyle \lambda$ and $\displaystyle \mu$:
$\displaystyle 18\mu - 4\lamda + 5 = 0$ and $\displaystyle 18\mu + 11 \lambda -61 = 0$which give (if my working is correct):
$\displaystyle \lambda = \dfrac{33}{13}$ and $\displaystyle \mu = \dfrac{67}{234}$Is that what you make it?
Grandad
Yeah!! that's exactly what i get!
But from the answers I've been giving for this question, putting this back into the equation for each line doesn't give me the correct $\displaystyle \overline{OA}$ and $\displaystyle \overline{OB}$ vectors.
These are meant to be: $\displaystyle \overline{OA}=(-5i - 7.5j + k)$ and $\displaystyle \overline{OB}=(8i - j + 14k)$
Sudharaka:
I get to a point where 202 - (56/3)$\displaystyle \lambda^2$ + (3304/39)$\displaystyle \lambda$ + (1981/39) + 234.((26$\displaystyle \lambda^2$-118$\displaystyle \lambda$ - 70.75)/78)^2 - 14$\displaystyle \lambda$ = 0
that squared after the bracket makes me think I will get $\displaystyle \lambda^4$ terms although I haven't expanded on it yet.
Hello maggiecThe answer you've been given is definitely incorrect. The point A doesn't lie on the line $\displaystyle L_1$.
With $\displaystyle \lambda = -1$, the point $\displaystyle -5\vec i -7.5 \vec j + \vec k$ lies on the line
$\displaystyle \vec r = -2\vec i +11.5 \vec j +\lambda(3\vec i + 4 \vec j - \vec k)$.I suspect a typo somewhere.
Grandad