Vector Simplification

[Csling]

I'd appreciate help with a vector simplification. Thanks in advance.

If |a| = |b|, simplify (a+b)^2.

It would be nice if the answer was explained, as I know the answer, I just don't know how to get to it in well presented manner, or, well, any manner.

Also, my bad if this is the wrong category for this question.

 

[HallsofIvy]

I'd appreciate help with a vector simplification. Thanks in advance.

If |a| = |b|, simplify (a+b)^2.

It would be nice if the answer was explained, as I know the answer, I just don't know how to get to it in well presented manner, or, well, any manner.

Also, my bad if this is the wrong category for this question.

Frankly the question makes no sense. There is NO multiplication defined for general vector spaces. For "inner product spaces", the dot product is defined. For $\displaystyle R^3$, both the dot product and the cross product. In any case "$\displaystyle (a+ b)^2$" is either undefined or ambiguous. Do you mean, perhaps, $\displaystyle |a+ b|^2$, the square of the length of the sums? That would be the same as $\displaystyle (a+ b)\cdot(a+ b)$ the dot product of a+ b with itself.

If so, the multiply out $\displaystyle (a+ b)\cdot (a+ b)$ according to the usual rules of algebra- the distributive law, and commutative law: $\displaystyle (a+ b)\cdot (a+ b)= a\cdot a+ a\cdot b+ b\cdot a+ b\cdot b$.

 

[Csling]

Unless it was misprinted in the textbook, it's definitely (a+b)^2. If it helps, it comes out to 2(a^2)(1+cos(θ))

 

[Deleted member 4993]

Unless it was misprinted in the textbook, it's definitely (a+b)^2. If it helps, it comes out to 2(a^2)(1+cos(θ))

So it is |a + b|²

|a + b|² = |a|² + |b|² + 2|a||b|cos(x)

when |a = |b|

|a + b|² = |a|² + |a|² +2|a|²cos(x) = 2|a|²[1+cos(x)]

 

[Csling]

Alright, thanks, sorry for the confusion.

 

[pka]

Unless it was misprinted in the textbook, it's definitely (a+b)^2. If it helps, it comes out to 2(a^2)(1+cos(θ))

I have no doubt that you copied the notation correctly. Over the last twenty or so years several books have said that vector multiplication is dot product. While it is wrong, in some cases it works out nicely. As in this: $\displaystyle (\vec{\bf{a}}+\vec{\bf{b}})^2=\vec{ \bf{a}}\cdot\vec{\bf{a}}+2\vec{ \bf{a}}\cdot\vec{\bf{b}}+\vec{\bf{b}}\cdot\vec{\bf{b}}$

Because $\displaystyle \|\vec{\bf{a}}\|=\|\vec{\bf{b}}\|~\&~\|\vec{\bf{a}}\|^2=\vec{\bf{a}}\cdot\vec{\bf{a}}$ then $\displaystyle (\vec{\bf{a}}+\vec{\bf{b}})^2=2\|\vec{ \bf{a}}\|^2+2(\vec{ \bf{a}}\cdot\vec{\bf{b}})$