1. ## Linear Algebra

Hi there,

I'm finding my algebra unit quite difficult compared to calculus... Could someone help with the below problem?

Let m, n and o be three vectors in R3 such that
m + n = o.
(a) Show that m x n = o x n = m x o.
(b) Use the geometric meaning of vector addition and cross
product to explain the statement of (a) geometrically. Make
sure that all the cases are covered.

2. ## Re: Linear Algebra

Let $\displaystyle m=(m_1,m_2,m_3), n = (n_1,n_2,n_3), o = (o_1,o_2,o_3)$. Calculate each cross product. Since $\displaystyle m+n=o$, you know $\displaystyle o = (m_1+n_1,m_2+n_2,m_3+n_3)$.

3. ## Re: Linear Algebra

You can also do this without considering coordinates. Using the fact that the cross product is linear in each argument and $a\times a=0$, multiply $m+n=o$ by $m$ on the left and by $n$ on the right.

4. ## Re: Linear Algebra

Thanks guys! Can anybody help with part b? I know the geometric meaning of cross product is |v x w| = |v||w|sin theta, but I'm not sure how to approach the problem

5. ## Re: Linear Algebra

I wouldn't consider that a "geometrical" meaning. Rather, the geometrical meaning of the cross product of two vectors is the area of the parallelogram having the two vectors as sides.

6. ## Re: Linear Algebra

It is also the vector that is normal (or perpendicular) to the plane containing the two vectors. The vector $m+n$ is in the same plane as both $m$ and $n$, so the same vector that is perpendicular to both $m$ and $n$ will also be perpendicular with $m$ and $m+n$ and $n$ and $m+n$.