Finding magnitude of a 3D vector with components in i, j, k notation

[Gattz]

If you are given vector a = a_xi + a_yj + a_zk and you want to find the magnitude of a, then would you just do the square root of a_x² + a_y²? If so, then what about a_z? What is a_zk actually? I'm a little unclear about it being related to a.

 

[Delphi51]

You would do the square root of ax2 + ay2 + ay2.
Sorry, I don't know how to make superscripts. Too bad they don't copy.

 

[Gattz]

You mean a_z for the last one?

And to do superscripts, just put "[.SUP] [/SUP.]" without the quotations and periods (caps don't matter). And it's _{for subscripts.}

 

[Delphi51]

Yes a_z, and thanks for the tip.

 

[Hannisch]

Just a quick explanation for that a_zk term, if you still are unclear about it.

k adds another dimension to the vector. When a vector is only in i,j it's 2D and i,j,k is 3D. See it as drawing it in a xyz-coordinate system instead of a xy one. And as was mentioned earlier, it changes things a little bit when it comes to the magnitude. You wouldn't just do the √(a_x² + a_y²), because then you wouldn't take the third dimension into consideration. From there it's not too big of a leap to √(a_x² + a_y² + a_z²).

You can prove it using Pythagoras' theroem. If you draw a 3D vector in a coordinate system you can figure it out. It can be a bit tricky to see, but it's not impossible.

 

[Gattz]

Is it something like http://upload.wikimedia.org/wikipedia/commons/thumb/f/fd/3D_Vector.svg/555px-3D_Vector.svg.png ?

So if it is then wouldn't you have to find the hypotenuse of a_z² + a_y² then take that length and find the hypotenuse of that and a_z?

 

[Delphi51]

You could do it that way and you'll get the same answer.
It would be interesting to prove that . . .

 

[Gattz]

Actually I just realized that the square root of a_z² + a_y² is the hypothesis. So I guess it makes sense.