# Inner Product

Last two inner product questions.

The first one I am little confused on and the second one I don't know what to do.

See worked attached.

#### Attachments

• 54.7 KB Views: 163

vela
Staff Emeritus
Homework Helper

I don't know how to do it so I am confused on what to do.

vela
Staff Emeritus
Homework Helper
If two vectors are orthogonal, what does that mean about their inner product? And what does it mean to say a vector is a unit vector?

Mark44
Mentor
Last two inner product questions.

The first one I am little confused on and the second one I don't know what to do.

See worked attached.
For the first one, show that <cos(mx), sin(nx)> = 0. Also show that <cos(mx), cos(mx)> = 1 and that <sin(nx), sin(nx)> = 1.

For the second one, you're showing that the "1 norm" is a norm on Rn. Verify that the formula satisfies all the defining properties of a norm: positive definiteness, etc. BTW, this norm simply adds the absolute values of the coordinates of a vector. For example, in R3, ||<2, -1, 3>||1 = |2| + |-1| + |3| = 6.

How can I integrate cos(mx)*sin(nx) when they don't have the same angle?

vela
Staff Emeritus
Homework Helper
Use a trig identity to rewrite the product of a sine and a cosine.

For the second problem, I don't know how to start the proof.

Mark44
Mentor
Integration by parts twice would probably work, or you could look in a table of integrals. Since this isn't about learning to integrate, but is instead and application of integration, it seems reasonable to me to look it up in a table of integrals.

On the integration by parts comment, I have an unrelated issue. I was doing a Laplace Transform of e^(-st+t)*sin(t) but after doing integration by parts, I had a form of the original with a 1/(s-1) time the integral so I cant subtract across to the other side. What can I do there since if I keep going it will an endless cyclical cycle?

I have attached a update that show cos(mx) and sin(nx) are orthogonal; however, showing cos(mx) cos(mx) equals 1 isn't quite working.

#### Attachments

• 157.6 KB Views: 95
vela
Staff Emeritus
Homework Helper
It didn't say it explicitly in the problem statement, but you need to assume m and n are integers. What is sine evaluated at any multiple of 2π?

If they are integers, then of course sin goes to zero.

Mark44
Mentor
On the integration by parts comment, I have an unrelated issue. I was doing a Laplace Transform of e^(-st+t)*sin(t) but after doing integration by parts, I had a form of the original with a 1/(s-1) time the integral so I cant subtract across to the other side. What can I do there since if I keep going it will an endless cyclical cycle?
Can you elaborate on this a bit more? You haven't said quite enough so that I'm not sure I understand what you're talking about. If you have a given integral on one side, and 1/(s - 1) times the same integral on the other side, add -1/(s - 1) times the integral to both sides, and then combine the two integrals using the rules of plain old fractions. At that point you can solve for the integral algebraically.