Question

I am having a little trouble understanding the inner product. Can someone please help me?

I have V=R^3
and (a1,b1,c1)^t * (a2,b2,c2)= 2a1a2 +3b1b2 + c1c2

I would like to know the length of (2,1,1)^t


I know that the length is equal to sqrt(v*v)

how can I find the length though, I am only given one vector. Don't I need 2 vectors, so that I can choose the first one to be a1,b1,c1, and the other one to be the other vector?

Thanks!

Answer 1

the product in this case is what you wrote, so you compute \((2,1,1)\cdot(2,1,1)\) and then take the root

Answer 2

i.e. \(2\times 2\times 2+3\times 1\times 1+1\times 1\)

Answer 3

unless i miss my guess the answer is \(\sqrt{12}\)

Answer 4

Ah. So what you are saying is that this is an inner product formula for any two vectors. You can get around that by just plugging in the vector twice.

So in my case,

I would have (2,1,1)

so the * of that would be (2*2*2) + (3*1*1) + (1*1)

= 12

so then the length would be \[\sqrt{12^2}\]

which is 12


is that what you mean?

Answer 5

not
\[\sqrt{12}^2\] but rather
\[\sqrt{12}\]

Answer 6

it is
\[\sqrt{\overrightarrow{v}\cdot{\overrightarrow{v}}}\]

Answer 7

ah! of course. The * isn't multiplication, it is the inner product.

that would make sense.

Thanks!

Answer 8

So using that same logic then

if we have A=(1,-1,1) and B=(2,1,1)

we can determine that they are NOT orthogonal becaues

A*B= (2*1*2) + (3*-1*1) + (1*1) = 2

and 2 doesn't equal 0, so therefore they can't be orthogonal.

Is that correct?