Unit Vector Confusion: Understanding Direction & Scalar Product Formula

[Rudders]

Hi,

I have a little confusion with vectors. I have memorised the formula so I can apply it, but I don't feel I really understand how it works.

One thing that confuses me is this. How does a unit vector exactly show direction. I've memorized and can blindly believe that this is how it is, but I was wondering if someone could show how it does this, then this'll hopefully enable me to understand and apply them better.

Also, if possible. I was wondering how the scalar product formula works (I think this has to do with the unit vector, but I'm not too sure).

The formula is: a.b = |a| |b| cos (theta)

Which finds the angle between two vectors... but I'm not sure how :(

Thanks heaps!
-Rudders

 

[EnumaElish]

http://mathworld.wolfram.com/Direction.html :
"The direction from an object A to another object B can be specified as a vector v = AB with tail at A and head at B. However, since this vector has length equal to the distance between the objects in addition to encoding the direction from the first to the second, it is natural to instead consider the unit vector v (sometimes called the direction vector), which decouples the distance from the direction."

Simply put: X.Y = |X| |Y| cos θ implies (X/|X|).(Y/|Y|) = cos θ. (Each of X/|X| and Y/|Y| is a unit vector.)

This also explains how the angle θ relates to unit vectors.

 

[tiny-tim]

Hi Rudders!

… I have memorised the formula so I can apply it, but I don't feel I really understand how it works.

Which formula are you talking about?

 

[Jame]

I think he means $$\frac{\vec u}{|u|}$$.

A unit vector is as the name tells a vector of unit length, i.e. its length is 1. As you can see by the Pythagorean theorem, there are many right triangles with a hypotenuse of length 1, and all of these can be seen as unit vectors where the length of the two other sides represent the x and y coordinates. Try this, draw a coordinate system on a paper, draw a vector arrow in some direction starting from the origin. Observe that you can shrink the arrow in length by moving it closer of away from the origin, where each length corresponds to x and y coordinates. This process is the geometric way of multiplying the vector with a number $$\lambda$$, and if this $$\lambda$$ happens to be $$1/|u|$$, I think you will see that this is analgous to multiplying a number a by 1/a, which gives 1.

 

[HallsofIvy]

Every vector has "length" and "direction". Multiplying or dividing a vector by a number changes the length but not the direction.

The confusion may be that you are thinking that the unit vector $$\frac{\vec{u}}{|\vec{u}|}$$ shows direction in some way that $$\vec{u}$$ itself doesn't. That is not true- they both show the same direction.

We prefer to use unit vectors to "show direction" in that they do not have "distracting" other information- length.

 

[Rudders]

Hmm. Think I understand now. Thanks :)