# Thread: Find the vector 'b'

1. ## Find the vector 'b'

Question :

Find a vector b such that a.b = 1 , a $\displaystyle \times$ b = j - k and a = i + j + k

2. Originally Posted by zorro
Question :

Find a vector b such that a.b = 1 , a $\displaystyle \times$ b = j - k and a = i + j + k
Let b = ui + vj + wk.

Substitute it and a = i + j + k into a.b = 1 and a $\displaystyle \times$ b = j - k.

Get some equations and use them to solve for u, v and w.

If you need more help, please show all of your working and state specifically where you're still stuck.

3. ## I am so sorry but i am stull stuck there

Originally Posted by mr fantastic
Let b = ui + vj + wk.

Substitute it and a = i + j + k into a.b = 1 and a $\displaystyle \times$ b = j - k.

Get some equations and use them to solve for u, v and w.

If you need more help, please show all of your working and state specifically where you're still stuck.

I am so sorry but i am still stuck there i tried to do what u adviced me to do but it s not leading me anywhere

This is what i have done till now!!!

$\displaystyle b = ui + vj + wk$ ; $\displaystyle a = i + j + k$

$\displaystyle a.b = 1$ and $\displaystyle a \times b = j - k$

$\displaystyle a.b = (i + j + k)(ui + vj + wk) = ui^2 + vj^2 + wk^2$

Now what should i do with $\displaystyle a.b$

And
$\displaystyle a \times b = \begin{vmatrix} i & j & k \\ 1 & 1 & 1 \\ u & v & w \end{vmatrix}$ = $\displaystyle i(v - w) - j(w - u) + k(v - u)$

therefore
$\displaystyle i(v - w) - j(w - u) + k(v - u) = j - k$

what should i do with this???

4. Originally Posted by zorro
I am so sorry but i am still stuck there i tried to do what u adviced me to do but it s not leading me anywhere

This is what i have done till now!!!

$\displaystyle b = ui + vj + wk$ ; $\displaystyle a = i + j + k$

$\displaystyle a.b = 1$ and $\displaystyle a \times b = j - k$

$\displaystyle a.b = (i + j + k)(ui + vj + wk) = ui^2 + vj^2 + wk^2$ Mr F says: You're expected to know that i^2 = j^2 = k^2 = 1. Therefore ....

Now what should i do with $\displaystyle a.b$

And
$\displaystyle a \times b = \begin{vmatrix} i & j & k \\ 1 & 1 & 1 \\ u & v & w \end{vmatrix}$ = $\displaystyle i(v - w) - j(w - u) + k(v - u)$

therefore
$\displaystyle i(v - w) - j(w - u) + k(v - u) = j - k$ Mr F says: Equate components on each side. eg. Equating i components gives v - w = 0.

what should i do with this???
Use the resulting four equations to solve for u, v and w (and hence b).

5. ## Is the answer correct ?

Originally Posted by mr fantastic
Use the resulting four equations to solve for u, v and w (and hence b).

Is this correct ?

$\displaystyle a.b = u + v + w$

And since
Mr F says: Equate components on each side. eg. Equating i components gives v - w = 0.

Therefore
$\displaystyle (v - w) = 0$
$\displaystyle (w - u) = -1$
$\displaystyle (v - u) = -1$

6. Originally Posted by zorro
Is this correct ?

$\displaystyle a.b = u + v + w$

And since
Mr F says: Equate components on each side. eg. Equating i components gives v - w = 0.

Therefore
$\displaystyle (v - w) = 0$
$\displaystyle (w - u) = -1$
$\displaystyle (v - u) = -1$
Regarding $\displaystyle a.b = u + v + w$, recall what the question said that a.b was equal to. Therefore ....

7. ## Is this correct?

Originally Posted by mr fantastic
Regarding $\displaystyle a.b = u + v + w$, recall what the question said that a.b was equal to. Therefore ....
ok

$\displaystyle u + v + w = 1$

Mr fantastic is this the right step or no

$\displaystyle u = 1 - v - w$

Substituting $\displaystyle u$ in $\displaystyle (v - u) = -1$ we get

$\displaystyle v - 1 + v + w = -1$
$\displaystyle 2v + w = 0$
$\displaystyle v = \frac{-w}{2}$

Substituting $\displaystyle v$ in $\displaystyle (v - w) = 0$ weget

$\displaystyle \frac{-w}{2} - w = 0$

$\displaystyle \frac{-w - 2w}{2} = 0$

$\displaystyle \frac{-3w}{2} = 0$

$\displaystyle w = - \frac{2}{3}$

substituting $\displaystyle w$ in $\displaystyle (v - w) = 0$

$\displaystyle v - \frac{2}{3} = 0$

$\displaystyle v = \frac{2}{3}$

Now Subtitute $\displaystyle v$ in $\displaystyle (v - u) = 0$

$\displaystyle \frac{2}{3} - u = 0$

$\displaystyle u = \frac{2}{3}$

Am i doing it correctly or no ?????

8. Originally Posted by zorro
ok

$\displaystyle u + v + w = 1$

Mr fantastic is this the right step or no

$\displaystyle u = 1 - v - w$

Substituting $\displaystyle u$ in $\displaystyle (v - u) = -1$ we get

$\displaystyle v - 1 + v + w = -1$
$\displaystyle 2v + w = 0$
$\displaystyle v = \frac{-w}{2}$

Substituting $\displaystyle v$ in $\displaystyle (v - w) = 0$ weget

$\displaystyle \frac{-w}{2} - w = 0$

$\displaystyle \frac{-w - 2w}{2} = 0$

$\displaystyle \frac{-3w}{2} = 0$

$\displaystyle w = - \frac{2}{3}$

substituting $\displaystyle w$ in $\displaystyle (v - w) = 0$

$\displaystyle v - \frac{2}{3} = 0$

$\displaystyle v = \frac{2}{3}$

Now Subtitute $\displaystyle v$ in $\displaystyle (v - u) = 0$

$\displaystyle \frac{2}{3} - u = 0$

$\displaystyle u = \frac{2}{3}$

Am i doing it correctly or no ?????
Have you checked your solution by substituting it into the equations?

From the equations: v = w, u = 1 + v = 1 + w.

Substitute into u + v + w = 1. Therefore ....

9. ## Is this correct

Originally Posted by mr fantastic
Have you checked your solution by substituting it into the equations?

From the equations: v = w, u = 1 + v = 1 + w.

Substitute into u + v + w = 1. Therefore ....

I am getting
$\displaystyle u = 1$
$\displaystyle v = 0$
$\displaystyle w = 0$

10. Originally Posted by zorro
I am getting
$\displaystyle u = 1$
$\displaystyle v = 0$
$\displaystyle w = 0$
Correct.

11. ## Thanks u every one for helping me

Thank u mr fantastic and every one else for helping me