Question Number 207812 by Davidtim last updated on 27/May/24
$${prove}\:{that}\:\frac{{vector}}{{scalar}}={vector} \\ $$
Answered by A5T last updated on 27/May/24
$${Let}\:{scalar}=\lambda\in\mathbb{R};\:{and}\:{vector},\boldsymbol{{a}}=\left({a}_{\mathrm{1}} ,{a}_{\mathrm{2}} ,…,{a}_{{n}} \right)\in\mathbb{R}_{{n}} \\ $$$${where}\:{each}\:{a}_{{i}} \in\mathbb{R}.\:{By}\:{definition}, \\ $$$$\frac{\mathrm{1}}{\lambda}\boldsymbol{{a}}=\left(\frac{{a}_{\mathrm{1}} }{\lambda},\frac{{a}_{\mathrm{2}} }{\lambda},…,\frac{{a}_{{n}} }{\lambda}\right).\:{Since}\:{each}\:\frac{{a}_{{i}} }{\lambda}\:{also}\:\in\mathbb{R},\frac{\boldsymbol{{a}}}{\lambda}\:{is} \\ $$$${a}\:{vector}. \\ $$
Commented by Davidtim last updated on 26/Oct/24
$${but}\:{when}\:{we}\:{find}\:{the}\:{work}\:{we}\:{follow}\:{this} \\ $$$${W}={F}\centerdot{d} \\ $$$${why}\:{here}\:{the}\:{work}\:{is}\:{scalar}? \\ $$
Commented by A5T last updated on 26/Oct/24
$${The}\:{mathematical}\:{idea}\:{of}\:{vectors}/{scalars}\:{is} \\ $$$${different}\:{from}\:{that}\:{in}\:{physics}. \\ $$