Using reduced row echelon to show if a vector is in a span

Question

Let $v_1,...,v_k$ be points in $\mathbb{R}^3$ and suppose that for $j = 1,...,k$ an object with mass $m_j$ is located at point $v_j$ . Physicists call such objects point masses. The total mass of the system of point masses is $m=m_1+...+m_k$. The center of gravity (or center of mass) of the system is

$\vec{v}=\frac{1}{m}[m_1v_1+...+m_kv_k]$.

Compute the center of gravity of the system consisting of the following point masses (see the figure):

$v_1 = (5, -4, 3); m_1=2g$

$v_2 = (4, 3, -2); m_2=5g$

$v_3 = (-4, -3, -1); m_3=2g$

$v_4 = (9, 8, 6); m_4=1g$

Let $v$ be the center of mass of a system of point masses located at $v_1,...v_k$. Is $v$ in $Span\{v_1,...,v_k\}$? Explain.

I calculated the center of mass which came out to be (1.3, 0.9, 0). I tried solving the stated question by creating the augmented matrix:

$ \begin{bmatrix} 5 & 4 & -4 & -9 & 1.3\\ 4 & 3 & -3 & 8 & 0.9\\ -3 & -2 & -1 & 6 & 0 \end{bmatrix} $

I then tried putting this into reduced echelon form, which did show that this matrix has infinite many solutions so that the center of mass is in the span. I was wondering if this was the correct approach? Because it was very cumbersome to calculate the reduced echelon form with this matrix.

Answer 1

Row-reducing an augmented matrix as you did is a perfectly good way to determine whether or not a specific vector is in the span of a set of vectors. However, there’s no need to do that here. $\operatorname{Span}\{v_1,\dots,v_k\}$ is the set of all linear combinations of the given vectors. By definition, $v=\frac1m[m_1v_1+\cdots+m_kv_k]$, which is a linear combination of the vectors $\{v_1,\dots,v_k\}$, so $v$ must be an element of their span.