dot product help?

Question

i seem to be failing miserably at this homework assignment. i need to prove that for two unit vectors E1 and E2 which are orthogonal to one another, for all vectors X, it is possible to express X as

X = (X . E1)E1 + (X . E2)E2

so, by defining the following

X = (x1, x2)
E1 = (e1, e2)
E2 = (e3, e4)

it follows that

e1e3 + e2e4 = 0
e1^2 + e2^2 = 1
e3^2 + e4^2 = 1

using these definitions, it follows that

(X . E1)E1 + (X . E2)E2 = ( x1(e1^2 + e3^2) + x2(e1e2 + e3e4), x1(e1e2 + e3e4) + x2(e2^2 + e4^2) )

and i can't reduce this to (x1, x2) because i don't see why

e1e2 + e3e4 = 0

and

e1^2 + e3^2 = e2^2 + e4^2 = 1.

hopefully someone will be able to help me, i'm really struggling with this. thanks a lot.

Answer 1

Hi,

My brain works better with numbers, so I decided E1=(3/5,4/5) and E2=(4/5,-3/5). You can see those are unit vectors and with slopes of 4/3 and -3/4, they are clearly perpendicular. I also left vector X = (7,2). Plugging these numbers into your equation, watch what happens.

( x1(e1^2 + e3^2) + x2(e1e2 + e3e4),
___ x1(e1e2 + e3e4) + x2(e2^2 + e4^2) )=

[7[(3/5)^2 + (4/5)^2] +2[ 3/5*4/5 + 4/5*(-3/5)],
___7[ 3/5*4/5 + 4/5*(-3/5)]+2[(4/5)^2+(-3/5)^2]=

[7(9/25 + 16/25]+2[12/25-12/25], 7[12/25-12/25]+2[16/25+9/25]=

7(1)+2(0), 7(0)+2(1)= 7+0,0+2=7,2

I hope this helps you to see more clearly how this works!

Answer 2

It is easy to prove in 3 lines , If a different approach is followed :
let X = aE1 + bE2

X.E1 = a E1.E1 + bE2.E1
X.E1 = a (1) + b(0) => a = X.E1
X.E2 = a(0) + b(1) => b = X.E2

so X = (X . E1)E1 + (X . E2)E2

Answer 3

I'm sorry I can't follow your notation, but are you aware that the dot product of orthogonal vectors is zero?

Answer 4

remember your law of cosines.
(a, b) = |a| |b| cos(theta)

thus, if they are orthogonal, your cos(theta) = 0

Answer 5

UH...ummmm..hmm..uhhh..Oh wait..no..hmmm damn...never mind I don't know. well wait, uhhh...9 no...78..no AHHHHHH I DONT EFFING KNOW!