1. ## Vectors

What can you conclude about a and b given that

(a) $\displaystyle ||a||^2+||b||^2=||a+b||^2$

(b)$\displaystyle ||a||^2+||b||^2=||a-b||^2$

I just do not like these open ended questions...I am never sure where to begin.

2. Recall u*u = ||u||^2

||a +b|| ^2 = (a+b)*(a+b) = ||a||^2 + 2a*b +||b||^2

what does this tell you?

3. One of the vectors is 0?

4. no a*b = 0

what does this tell you ?

5. Ohhh, they are perpendicular?

(How does the second problem differ?)

6. you now know to answer that yourself.

also it woud be a good time to reconsider what Captain Black had to say

7. $\displaystyle ||a-b||^2=(a-b)(a-b)=||a||^2-2ab+||b||^2$

I'm not sure how this differs from part (a) in that it just perpendicular vectors.

I wasn't very clear on what captain black said about c=-b.

8. remember a - b is a + (-b)

think about it --- consider the vectors i + j and i - j

what is the difference between these 2?

9. One vector is positive and one is negative?

$\displaystyle ||a+(-b)||^2=(a+(-b))(a+(-b))=a^2+2a(-b)+(-b)^2$

10. I'm not trying to sound rude but I have no idea what you are trying to say.

11. Originally Posted by Zocken
One vector is positive and one is negative?

$\displaystyle ||a+(-b)||^2=(a+(-b))(a+(-b))=a^2+2a(-b)+(-b)^2$
When $\displaystyle c=-b$

$\displaystyle ||a-b||^2=||a+c||^2$

and $\displaystyle ||c||=||b||$

So now we are asking: What can you conclude when:

$\displaystyle ||a+c||^2=||a||^2+||c||^2$

and then what does that mean for $\displaystyle a$ and $\displaystyle b$

12. Does it mean b=0?

or a>b?

13. Does it mean b=0?

or a>b?

first of all vectors cannot be positive or negative. and it makes no sense

to say things like a > b when and b are vectors.

secondly the point i wanted you to see is that
1. for both i + j and i - j |i+j|^2 = |i|^2 +|j|^2
2. i an j are perpindicular
3. If you graph the vectors notice that i , j and i+j form a right triangle

as does i,j, and i - j which is the point Captain Black was making when he asked you to consider a parallelogram

I think we were both trrying to get you to realize on your own with hints

If
a)

(b)

then a and b are perpindicular period.