Let \(v\) be the center of mass of a system of point masses located at \({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_k}\) as in exercise 29. Is \(v\) in Span \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_k}} \right\}\)? Explain.

It is given that \({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_k}\) are points in \({\mathbb{R}^3}\) and \(v\) is the center of mass of a system of point masses located at \({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_k}\).

The center of gravity (or center of mass) of the system is given as:

\(\bar v = \frac{1}{m}\left[ {{m_1}{v_1} + {m_2}{v_2} + ... + {m_k}{v_k}} \right]\)

Thescalar multiple of a vector\({\mathop{\rm u}\nolimits} \)by real number\(c\)is the vector\(c{\mathop{\rm u}\nolimits} \)obtained by multiplying each entry in\({\mathop{\rm u}\nolimits} \)by\(c\).

The center of gravity of the system is:

\(\bar v = \frac{{{m_1}}}{m}{v_1} + \frac{{{m_2}}}{m}{v_2} + ... + \frac{{{m_k}}}{m}{v_k}\)

If \({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}\) is in \({\mathbb{R}^n}\), then the set of all linear combinations \({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}\) is denoted by span \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}} \right\}\), and is called the subset of \({\mathbb{R}^n}\) spanned \({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}\). A span is a collection of all vectors that can be written in the form of \({c_1}{v_1} + {c_2}{v_2} + .... + {c_p}{v_p}\).

The obtained expression of the center of gravity of the system represents \({\mathop{\rm v}\nolimits} \) as a linear combination of \({v_1},...,{v_k}\), which indicates that \(v\) is in \({\mathop{\rm Span}\nolimits} \left\{ {{v_1},...,{v_k}} \right\}\).