1. ## Vectors

I am doing part c) of the following question and was wondering if anyone can tell me what they get for position vectors of A and B as I'm not sure mine are correct.

Thanks!

2. Originally Posted by maggiec
I am doing part c) of the following question and was wondering if anyone can tell me what they get for position vectors of A and B as I'm not sure mine are correct.

Thanks!
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.

3. Originally Posted by Sudharaka
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.

4. Originally Posted by maggiec
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)

5. Originally Posted by Sudharaka
Dear maggiec,
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: $\displaystyle (13.5+7mu-3lambda)i + (-8.5+8mu+4lambda)j + (8.5-11mu+lambda)k$

6. Hello maggiec
Originally Posted by maggiec
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: $\displaystyle (13.5+7mu-3lambda)i + (-8.5+8mu+4lambda)j + (8.5-11mu+lambda)k$
I 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?

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?

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.

8. Hello maggiec
Originally Posted by maggiec
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.
The 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.

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.

Ok, I found the typo. I am attaching the answer I've been given and I have circled the mistake in red:

So this means the correct answer is as follows? ;

$\displaystyle \overline{OA}=((73/13)i + (35/26)j - (33/13)k)$

and

$\displaystyle \overline{OB}=((1580/117)i + (619/117)j + (626/117)k)$

?

Thanks

10. Hello maggiec
Originally Posted by maggiec
Ok, I found the typo. I am attaching the answer I've been given and I have circled the mistake in red:

So this means the correct answer is as follows? ;

$\displaystyle \overline{OA}=((73/13)i + (35/26)j - (33/13)k)$

and

$\displaystyle \overline{OB}=((1580/117)i + (619/117)j + (626/117)k)$

?

Thanks