What's the Difference Between Ax and i-Hat in Vector Notation?

[jdawg]

Homework Statement

So this isn't really a specific homework question, it's more of a general one. What is the difference between a_x and i(hat)? I thought they were the same thing. Can someone please explain the difference?

Homework Equations

The Attempt at a Solution

 

[Chestermiller]

By a_x, are you referring to a coordinate basis vector? Please provide more context.

 

[jdawg]

Sorry, yes that's what I meant. Like the x component of a vector a.

 

[Chestermiller]

Sorry, yes that's what I meant. Like the x component of a vector a.

It still isn't clear to me. Please provide a specific example or two.

Chet

 

[jdawg]

Hmm... I don't know if I understand this well enough to even formulate a proper question... In my textbook it says vector a=a_x*i(hat)+a_y*j(hat). I get that the x component and the y component make up vector a, but I'm confused about the i(hat) and j(hat). I don't understand why the i and j are there :(

 

[BvU]

i(hat) is commonly used as the unit vector in the x direction
j(hat) idem in the y direction.

So if for example vector a=[5,3] they mean 5 units in the x-direction and 3 units in the y direction, in other words 5 * i(hat) + 3 * j(hat).

a is then the sum of two vectors: one is 5* i(hat), it points in the x-direction and has length 5, the second is 3* j(hat), it points in the y-direction and has length 3

the length of the x-component of a is then a_x = 5 and
the length of the y-component of a is then a_y = 3

Often, vectors are printed in bold, as I did with the a to distinguish them from numbers.
Because they are vectors too, I should also have used [itex]\bf\hat i[/itex] and [itex]\bf\hat j[/itex] or better, bold versions of [itex]\bf\hat \imath[/itex] and [itex]\bf\hat \jmath[/itex] without the dot, and perhaps upright but I can't find them...

 

[jdawg]

Oh ok, I kind of had a vague understanding that i(hat) was along the x axis, j along the y axis, and k along the z axis but I don't really get why they're there. Can't you tell that a component is on a certain axis by just looking at the sub letter? Like a_x is on the x axis, ect... So is using i the same as using a_x? By the way, where did you find the hat symbol? I don't see it on here.

 

[SteamKing]

It's different notations for different things. The i, j, k are unit vectors which are parallel to the x, y, and z axes, respectively. ax, ay, az can be the scalar magnitudes of the components of a vector, they can be component vectors, whatever. It depends on the particular usage. The magnitude of i, j, or k is always 1 (since they are unit vectors), while ax, ay, az can be anything.

 

[jdawg]

Ohhh so i, j, and k aren't actually on the axes?

 

[SteamKing]

The unit vectors i, j, and k have the following 3-D representation:

i = (1, 0, 0)
j = (0, 1, 0)
k = (0, 0, 1)

with the origin at (0, 0, 0)

These three vectors are mutually perpendicular to one another and are aligned and coincident with the three cartesian coordinate axes.

 

[Chestermiller]

Ohhh so i, j, and k aren't actually on the axes?

No. They are directions parallel to the axes. You express vectors as the sum of scalar components times the unit vectors so that you add and subtract the individual terms like vectors (so they automatically satisfy the parallelogram rule for adding and subtracting vectors). It makes it very convenient to handle them mathematically. In physics problems, you are often resolving vectors into their components perpendicular and tangent to surfaces. In such cases, the unit vectors perpendicular and tangent to the surfaces come in particularly handy, and they are often not parallel to the coordinate axes.

Chet