I need help with math
I have a formula which I know how to use, but am a little unsure of how it came to be and would appreciate if anyone with understanding would clarify this up for me.
[b]v[/b] = ll[b]v[/b]ll (cos(a)i, sin(a)j)
This is what I understand so far:
Let's say I draw a vector with a horizontal component of cos(a) and a vertical component of sin(a).
Since I know that magnitude is equal to sq.rt(x^2 + y^2), I substitute the values and get sin^2(a) + cos^2(a), which is a nice Pythagorean identity which equals one.
Since the magnitude of the vector equals 1, then the vector is a unit vector.
Furthermore, if I were to try to find the angle a of vector [b]v[/b], after substituting tan(a) = y/x, I notice that what ends up resulting is a simple identity of tangent.
However, I don't understand how someone went from this understanding to be able to say [b]v[/b] = ll[b]v[/b]ll (cos(a)i, sin(a)i).
To be clear, I understand what the formula is useful for. I also suppose that if I were to simply see it as a magnitude being multiplied by a horizontal and vertical component, then I can see how intuitively you can say it'll lead to a vector (a,b) or <a,b>, but I don't feel like that's solid enough. Does anyone know some proof to this formula?
Thanks for any input.
6 Comments • Newest first
You can always try http://www.khanacademy.org/
[quote=Clericase]I can't really prove it persay, I can explain why it works.
When you have a right triangle, and you have angle theta. The two legs of the triangle will equal:
Hypotenuse*cos(theta)
Hypotenuse*sin(theta)
In other words, the cos/sin will give you the simplest ratio between the two sides, you fit it to the triangle by multiplying by the hypotenuse. Conceptually, imagine say the 30 degree angle in a 30-60-90 triangle. The angle will be the same if the sides were 1, root 3, 2 or 5, 5 root 3, 10. (try it)
Now, for that equation to work, you have to be able to see the vector in its components and it'll form a right triangle. It's component along the x will be cos (theta) and the y would be sin (theta). Now taking the earlier concept into mind, the vector calculations are no different. In this case the magnitude lvl is the hypotenuse.
Hope that made some sense.[/quote]
This is perfect.
I had what you said vaguely illustrated in my mind, but it's much more clear now and I was unsure if I wasn't taking something else into account or something.
Thanks.
[quote=WeffTooFat]Dude, there really isn't a proof for this cuz it's not really much of a formula in the first place. Since you already understand that it's about breaking into horizontal and vertical components, all that's left is your sense of spacial thinking. Otherwise, this "formula" is the same as 2 = 1+1.
And proving 2 = 1+1 rigorously is pretty much impossible.[/quote]
I have an exam next week and I missed the last two days of lecture because I wasn't in-state, so since I've had to wing it by myself, I just wanted to make sure I understood.
I mean, yeah, if you think about it, it is kind of intuitive, but I was making sure I wasn't missing anything.
[quote=BrayishGlob]Youtube it, there's an online teacher that posts a video for every math lesson and goes into detail on them.[/quote]
Well, KhanAcademy doesn't mention what I'm looking for as far as I know, and I looked up PatrickJMT too and couldn't find specifically what I'm looking for.
Or are you talking about someone else?
[quote=Fairyic]Lol
Does it matter?
Just memorize the formula~[/quote]
Yes, it matters.
Lol
Does it matter?
Just memorize the formula~
[quote=Dexless4ever]you lost me at "The"[/quote]
Thanks for posting, now I can use your post to bump my thread.