Magnitude of vector, negative in part.

[feathermoon]

This is actually not a full problem, just a part of one I'm having trouble with:

Homework Statement

If I have a acceleration vector, say [ b[itex]k^{2}[/itex][itex]e^{kt}[/itex] - b[itex]c^{2}[/itex][itex]e^{kt}[/itex] ][itex]e_{r}[/itex] + [ 2bkc[itex]e^{kt}[/itex] ] [itex]e_{θ}[/itex]. How can I find its magnitude?

Homework Equations

Mag vector |a| = (a^2)^(1/2)

The Attempt at a Solution

As I square [itex]_{e}r[/itex], the cross term is still negative under the radical, and doesn't subtract cleanly:

[ [itex]b^{2}k^{4}e^{kt}[/itex] + [itex]b^{2}c^{4}e^{kt}[/itex] - 2[itex]b^{2}k^{2}c^{2}e^{kt}[/itex] ][itex]^{1/2}[/itex]

I'm either doing some wrong algebra or missing something obvious I think?

 

[feathermoon]

Well I'm stupid.. neglected e_theta completely out of sheer confusion from the first part... guess what it does when you add it in.

...Maybe this thread should be deleted. U_U Sometimes I just need a new perspective I guess.

 

[Redbelly98]

Glad it worked out. FYI, I have two comments to help you in the future:

1. Note that [itex](e^{kt})^2 = e^{2kt}, \text{ not } e^{kt}[/itex]

2. You can enclose an entire equation or large expression in itex-/itex tags, you do not need to use separate itex-/itex tags for selected terms.

So instead of

[ [itex]bk^{2}[/itex][itex]e^{kt}[/itex] - b[itex]c^{2}[/itex][itex]e^{kt}[/itex] ][itex]e_{r}[/itex] + [ 2bkc[itex]e^{kt}[/itex] ] [itex]e_{θ}[/itex]

you can write

[itex] [ bk^{2}e^{kt} - bc^{2}e^{kt} ] e_{r} + [ 2bkce^{kt} ] e_{θ} [/itex]

which gives you

[itex] [ bk^{2}e^{kt} - bc^{2}e^{kt} ] e_{r} + [ 2bkce^{kt} ] e_{θ} [/itex]

 

[feathermoon]

Thanks Redbelly! I did have e^kt as e^2kt in my calculation, I just forgot to transcribe that correctly.

The itex thing on the other hand will probably save me some time on my next questions! :D

 

[asteriatic]

It appears that you are trying to find the magnitude of the acceleration vector using the formula |a| = √(a^2), which is correct. However, your algebra may be incorrect. Let's take a closer look at the vector:

a = [ bk^{2}e^{kt} - bc^{2}e^{kt} ]e_{r} + [ 2bkce^{kt} ] e_{θ}

We can rewrite this as:

a = [ bk^{2}e^{kt} - bc^{2}e^{kt} ]e_{r} + [ 2bkce^{kt} ] e_{θ}

= [ bk^{2}e^{kt} - bc^{2}e^{kt} ](cosθ, sinθ) + [ 2bkce^{kt} ] (-sinθ, cosθ)

= [ bk^{2}e^{kt} - bc^{2}e^{kt} ]cosθ + [ 2bkce^{kt} ](-sinθ)

= bk^{2}e^{kt}cosθ - bc^{2}e^{kt}cosθ - 2bkce^{kt}sinθ

Now, we can use the formula for the magnitude:

|a| = √[(bk^{2}e^{kt}cosθ - bc^{2}e^{kt}cosθ - 2bkce^{kt}sinθ)^2]

= √(b^2k^4e^{2kt}cos^2θ - 2b^2k^2c^2e^{2kt}cosθsinθ + b^2c^4e^{2kt}cos^2θ + 4b^2k^2c^2e^{2kt}sin^2θ - 4b^2kce^{2kt}sinθcosθ)

= √(b^2k^4e^{2kt}cos^2θ + b^2c^4e^{2kt}cos^2θ + 4b^2k^2c^2e^{2kt}sin^2θ - 4b^2kce^{2kt}sinθcosθ)

= √(b^2k^4e^{2kt} + b^2c^4e^{2kt} + 4b^2