Simplifying (2u+v).(u-2v): A Step-by-Step Guide

In summary, to expand and simplify the expression (2u+v).(u-2v), you can use the distributive property and the dot product formula to get 2(u.u)- 3 u.v- 2(v.v). Alternatively, you can use the magnitude of vectors formula to get |u|2- 3u.v- 2|v|2.
  • #1
PiRsq
112
0
How do you expand and simplify this one?


(2u+v).(u-2v)
=2u.u+2u.(-2v)+v.u+v.(-2v)

Where u and v are vectors and the "." is the dot. I did some but after this I am lost, how can I continue?
 
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  • #2
I don't see what your problem is. You have done
(2u+v).(u-2v)=2u.u+2u.(-2v)+v.u+v.(-2v) correctly. Now just
combine 2u.(-2v)= -4 u.v and v.u which is also u.v:
This is 2(u.u)- 3 u.v- 2(v.v)

You can also use the fact that |u|= sqrt(u.u) so that
u.u= |u|2 and v.v= |v|2 so that
(2u+v).(u-2v)= |u|2- 3u.v- 2|v|2.
 
  • #3
Got it, thanks :smile:
 

What is the purpose of simplifying (2u+v).(u-2v)?

The purpose of simplifying (2u+v).(u-2v) is to reduce the expression to its simplest form by combining like terms and removing any unnecessary parentheses or symbols. This makes the expression easier to work with and can help solve equations or simplify further calculations.

What are the steps involved in simplifying (2u+v).(u-2v)?

The steps involved in simplifying (2u+v).(u-2v) are as follows:

  1. Distribute the first term (2u) to the second expression (u-2v).
  2. Distribute the second term (v) to the second expression (u-2v).
  3. Combine like terms, if any.
  4. Remove any unnecessary parentheses.

Can (2u+v).(u-2v) be simplified further?

Yes, (2u+v).(u-2v) can be simplified further if there are any like terms that can be combined or if any parentheses can be removed. However, the final simplified form may vary depending on the given expression.

What are the common mistakes to avoid when simplifying (2u+v).(u-2v)?

Some common mistakes to avoid when simplifying (2u+v).(u-2v) are:

  • Forgetting to distribute the first and second terms to the second expression.
  • Not combining like terms, if any.
  • Removing necessary parentheses.

Can simplifying (2u+v).(u-2v) be applied to other expressions?

Yes, the steps and principles used to simplify (2u+v).(u-2v) can be applied to other similar expressions involving multiplication and addition or subtraction of variables and constants.

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