Unit Vector Question

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January 31st 2009, 09:18 PM
Joeda

Unit Vector Question

Going back over textbook problems and having a bit of trouble with this one.
If v=ai+bj, show that a/((a^2+b^2)^1/2)=Cos theta & b/((a^2+b^2)^1/2)=Sin theta, where theta is the direction of v.
Sorry if the question is not written the best. Newbie to the site. Thanks for your help in advance.
January 31st 2009, 09:30 PM
Chris L T521

Quote:

Originally Posted by Joeda

Going back over textbook problems and having a bit of trouble with this one.
If v=ai+bj, show that a/((a^2+b^2)^1/2)=Cos theta & b/((a^2+b^2)^1/2)=Sin theta, where theta is the direction of v.
Sorry if the question is not written the best. Newbie to the site. Thanks for your help in advance.

I would suggest that you first draw a diagram.

The vector $\mathbf{v}=a\mathbf{i}+b\mathbf{j}$ has a component of length $a$ in the x direction, and a component of length $b$ in the y direction. The length of the hypotenuse of the triangle (which is the length of the vector) is given by $\parallel\!\mathbf{v}\!\parallel=\sqrt{a^2+b^2}$ .

Now, assign the angle between the vector and the positive x axis the value $\theta$ . Can you try to visualize what I described?

You should then be able to determine $\cos\theta$ and $\sin\theta$ .

Does this make sense?
January 31st 2009, 09:42 PM
Joeda

That does make sense thanks.
So to answer the question would I just have to draw the diagram and label all vectors?
January 31st 2009, 09:44 PM
Chris L T521

Quote:

Originally Posted by Joeda

That does make sense thanks.
So to answer the question would I just have to draw the diagram and label all vectors?

That isn't the answer itself. It leads you to the answer, which is to show that $\cos\theta=\frac{a}{\sqrt{a^2+b^2}}$ and $\sin\theta=\frac{b}{\sqrt{a^2+b^2}}$
January 31st 2009, 10:00 PM
Joeda

ok so i think ive got it. Im using {v} as my symbol for magnitude of v.

v=ai+bj
{v}=((a^2+b^2)^1/2)

and u=v/{v}
=a/((a^2+b^2)^1/2)

and we have this equatioin: u=(cos theta)i + (sin theta)j

so how do I put it all together to prove the question?
January 31st 2009, 10:03 PM
Chris L T521

Quote:

Originally Posted by Joeda

ok so i think ive got it. Im using {v} as my symbol for magnitude of v.

v=ai+bj
{v}=((a^2+b^2)^1/2)

and u=v/{v}
=(ai+bj)/((a^2+b^2)^1/2)

and we have this equatioin: u=(cos theta)i + (sin theta)j

so how do I put it all together to prove the question?

I inserted missing information in your post in red.

Now break it up into $\frac{a}{\sqrt{a^2+b^2}}\mathbf{i}+\frac{b}{\sqrt{ a^2+b^2}}\mathbf{j}$ and then the result follows.