# i hat, j hat, and k hat

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I just finished learning about vector components in my class, and I was hoping to understand the vector notation i hat, j hat, and k hat, but our book doesn't use that notation and gives an unsatisfying description of what the bold units stand for. Can someone please give me an overview of what they mean, and why they are without units? Thanks...

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It's funny - I just learned about these in class today too. They are basically the direction part of a vector. When coupled with a magnitude, they are a complete interpretation of the vector in that specific dimension. I think it went $\vec{A_x} = (A_x)\hat{i}$ Where the first Ax should have the vector sign over it and the i should have the hat (don't know how to do in latex.)

I think the i is unitless because it is an angle.

Edit: Yes Cap'n. Thanks for the latex help. (Edited latex)

Oh, and futhermore, $\hat{i}$ is for x, $\hat{j}$ for y, and $\hat{k}$ for z. A vector can be solved by $\vec{A}=(A_x)\hat{i}+(A_y)\hat{j}+(A_z)\hat{k}$ if I'm not mistaken.

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Something like this, you mean?

$\vec{A_x} = (A_x)\hat{i}$

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A vector can be solved by ...

"solved by" is definately the wrong word, "represented by" would be much more accurate.

One thing I would like to say is to not fall in love with i,j,&k, since many other books use other notations. Learn the concept, don't get hung up by the notation (you would not believe how much trouble some students have with this).

Really, those basis vectors (the bold letters) are there to help remind you which part of the vector points in each direction. That is, what part of any given vector points in the x direction, the y direction, etc. There are many different way of representing this: (let ~ mean "represented by")

v ~ [vx vy vz] ~ vxi + vyj + vzk

It is really a notation thing to aid in bookkeeping.

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Cool, thanks for the replies. If I'm getting this right it means that vector A sub x (sorry I'm horrible with latex) is simply the magnitude of the vector while i hat is just a unit of 1, but can be negative or positive to represent direction.

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Cool, thanks for the replies. If I'm getting this right it means that vector A sub x (sorry I'm horrible with latex) is simply the magnitude of the vector while i hat is just a unit of 1, but can be negative or positive to represent direction.

I think the $A_x$ also includes the angle since it is $A_x\cos{\theta}$ and the $\hat{i}$ is just $1$ or $-1$. I think anyway, I'll check my notes later tonight.

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Cool, thanks for the replies. If I'm getting this right it means that vector A sub x (sorry I'm horrible with latex) is simply the magnitude of the vector while i hat is just a unit of 1, but can be negative or positive to represent direction.

You have to very careful here to make sure your words say exactly what you are trying to say.

There is a phrase in here that you said was impossible: "...vector A_x is simply the magnitude..."

A vector cannot be the same things as a magnitude. One is a vector quantity, and the other is a scalar. They cannot be equal. The magnitude of a vector is a scalar. When writing equations, a vector can only be equal to another vector. The components A_x, A_y, etc. are scalars, but when you write them together with basis vectors, you are representing a vector then.

What the phrase should have been was "component A_x of the vector A is simply the magnitude..."

Also, i cannot be positive or negative. It is a basis vector that defines in what direction the x coordinte points. If you want a vector that points in the other direction, you put a negative component in front of it.

Also, the cosine terms comes from projection operators. They arise when given a vector's magnitude, how much goes along the x-coordinate. JustStuit's example should not have had a subscript x next to the A, just the magnitude of A. That is probably a lesson or two away, and really doesn't have anything to do with just using notation to describe a vector.

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• 4 weeks later...

Have you gone to your instructor after class and asked for help? I've always found that face-to-face exchanges work better, and on top of it he'll like that you're interested.

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$\hat i \hat j \hat k$ are the same as $\hat x \hat y \hat z$ They are unit vectors. Vectors of length 1 pointing in a direction which defines the axis of the coordinate system, so in cartesians would lie along the x,y,z axies. They are at right angles to each other.

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I find a few examples are often helpful, my physics class was flabbergasted about vector notation until tehy were put in more concrete numbers.

Does this help?

A particle moving 10 m/s north and 15 m/s east would have component notation like this: (10, 15) m/s,

or... (10i + 15j) m/s.

If the particle accelerates constantly north at 2 m/s^2, then the velocity is:

(10 + 2t, 15) m/s

or... ( (10 + 2t)i + 15j) m/s.

It's basically a conversion from the co-ordinate system into a single equation.

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• 6 years later...

My question has been, for a bloody while now, how do you write the in a way that you can actually move from place to place... a la Tex or Unicode or some special way of writing it as an equation in word... something.

Arrow indicators over letters (as attached) are similarly impossible to find. It's a real pain in the ass if you want to go all digital on your HW. You'd think by now it would be baked in to the various applications that allow you to write your own custom equations or deal with them (wolfram Alpha,Mathematica, Maple, SOMETHING). Frustratingly not the case. :/

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In LaTex you have \vect and \overrightarrow. Is that what you are asking about?

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