Question

consider the vector I =[ 1/Square root (2) , 1/Square root (2)] and J= [ -1/Square root (2), 1/Square root (2)]. Write the vector (2,-6) in terms of I and J.

Answer 1

james gave you a good starting point.
I= [1 1]/sqrt(2)
J= [-1 1]/sqrt(2)
you want to find two scalars (two simple numbers) that you multiply I and J by to get (2,-6)

call the numbers a and b
then you must solve
(2,-6)= a*I + b*J

Answer 2

First rule: multiplying a vector by a scalar uses this rule: a (x,y)= (ax, ay)

Answer 3

so, start with
(2,-6)= a*I + b*J
(2, -6) = a*(1,1)/sqrt(2) + b*(-1,1)/sqrt(2)

multiply both sides by sqrt(2) . This means multiply every term by sqrt(2)
you get
sqrt(2) * (2, -6) = a*(1,1) + b*(-1,1)

Answer 4

hmm this one doesn't make sense to me at all

Answer 5

ohh ok

Answer 6

Now use the rule
First rule: multiplying a vector by a scalar uses this rule: a (x,y)= (ax, ay)
on all the terms

Answer 7

so its 2 sqrt(2) , -6 sqrt(2) = a (1,1) + b (-1,1)

Answer 8

yes, but apply the rule to the two terms on the right hand side also

Answer 9

how do you do that?

Answer 10

a (x,y)= (ax, ay)

Answer 11

so its 2sqrt(2) , -6sqrt(2) = (a1, a1) + (-b, b1)

Answer 12

well, I would not write a1, because a*1 is just a.... also, put the left side in parens because we do not want to forget is is a vector

Answer 13

ok. its [2sqrt(2), -6sqrt(2)] = (a,a) + (-b,b) ?

Answer 14

yes, now add the two vectors on the right hand side

Answer 15

how do you do that?

Answer 16

You were able to subtract two vectors, for example P-Q in the last problem. Adding is the same, except you add instead of subtract.

Answer 17

so its like (1+-1, 1+1) = (0,2) ?

Answer 18

yes, but now we are adding variables instead of numbers

Answer 19

what do you mean?

Answer 20

(a,a) + (-b,b) = ??
add corresponding entries

Answer 21

(-ab, ab)

Answer 22

add (not multiply)

Answer 23

(a+-b, a+b)

Answer 24

yes, but normally you would write a + -b as just a-b .

Answer 25

oh ok. now what do i do?

Answer 26

You have
(2sqrt(2), -6sqrt(2)) = (a-b, a+b)

Answer 27

this means you have two vectors that equal each other. If they equal each other, what does that mean about each entry in the vector?

Answer 28

umm no clue

Answer 29

if I told you
( 1, 2) = (x, y)
what is x and what is y
how do you know?

Answer 30

that x = 1 and y =2

Answer 31

Now use the same idea on our problem
(2sqrt(2), -6sqrt(2)) = (a-b, a+b)

Answer 32

that a-b = 2sqrt(2) and a+b = -6sqrt(2)

Answer 33

yes. we have 2 equations and two unknowns. The way to solve these types of equations is

write them down like this:
a-b = 2sqrt(2)
a+b = -6sqrt(2)

add the two equations together (we are adding equal things to equal things, so the sum of them is still equal things)
2a-b+b= 2sqrt(2)-6sqrt(2)

can you solve for a?

Answer 34

lol thats confusing

Answer 35

would it be a = 2sqrt(2) -6sqrt(2) / 2 ??

Answer 36

first, you should put parens around (2sqrt(2) -6sqrt(2)) because you are dividing *both* terms by 2 (you divide the whole right hand side by 2)
next, you should simplify

Answer 37

so its [sqrt(2), -3sqrt(2)]?

Answer 38

it is not a vector.

Answer 39

what do you mean?

Answer 40

oh sqrt(2)i , -3sqrt(2) j

Answer 41

no. I mean you start with (2*sqrt(2)-6*sqrt(2))/2 which is just a (ugly) number:
divide both terms by 2 to get
(sqrt(2)-3*sqrt(2))
factor out sqrt(2) to get (1-3)*sqrt(2)
simplify 1-3 to get
a= -2*sqrt(2)

Answer 42

btw, if you want info on how to solve simultaneous equations like this one, watch
http://www.khanacademy.org/video/solving-systems-by-elimination?topic=worked-examples-4
and his other examples

Answer 43

ok

Answer 44

but what would be the answer for this?

Answer 45

we are getting there....
use
a+b = -6sqrt(2)
plus your value for a that we just found to get b

Answer 46

so b = -6sqrt(2) - 2sqrt(2) ?

Answer 47

well let's check that
a + b= -6sqrt(2)
-2*sqrt(2) + b= -6sqrt(2)

add +2sqrt(2) to both sides to get
b = -6sqrt(2)+2sqrt(2)
so close, but not exactly correct.
can you simplify this?

Answer 48

lol i am so confused

Answer 49

can you simplify
b = -6sqrt(2)+2sqrt(2)

Answer 50

umm -4sqrt(2) ?

Answer 51

yes.
we are almost done.

Answer 52

we have to answer
Write the vector (2,-6) in terms of I and J.

which we decided meant:
(2,-6)= a*I + b*J

we now know a and b, so the final answer is?

Answer 53

2sqrt(2) i + -4sqrt(2) j ?

Answer 54

and what is I and J ? It does not hurt to write out the whole solution

Answer 55

what do you mean?

Answer 56

I =[ 1/Square root (2) , 1/Square root (2)]
J= [ -1/Square root (2), 1/Square root (2)]
[2,-6]=2sqrt(2) I + -4sqrt(2) J

Answer 57

You can check your answer by multiplying the scalars times the vectors, and then adding the vectors to see if you get [2, -6]

btw, if you have time, you might want to brush up on your algebra using
http://www.khanacademy.org/#core-algebra

Answer 58

ohh k

Answer 59

thanks a lot