Question

If $\overline{a},\overline{b},\overline{c}$ are non-coplaner vectors, then prove that the vectors $3\overline{a}+\overline{b}+\overline{c},2\overline{a}+2\overline{b}+3\overline{c},\overline{a}+3\overline{b}+5\overline{c}$ are collinear.

Answer 1

$A(3\overline{a}+\overline{b}+\overline{c})B(2\overline{a}+2\overline{b}+3\overline{c})C(\overline{a}+3\overline{b}+5\overline{c})$

$AB+BC=AC$

$AB=\sqrt{(3-2{)}^2+(1-2{)}^2+(1-3{)}^2}=\sqrt{1+1+4}=\sqrt{6}$

$BC=\sqrt{(2-1{)}^2+(2-3{)}^2+(3-5{)}^2}=\sqrt{1+1+4}=\sqrt{6}$

$AC=\sqrt{(3-1{)}^2+(1-3{)}^2+(1-5{)}^2}=\sqrt{4+4+16}=\sqrt{24}=2\sqrt{6}$

$\sqrt{6}+\sqrt{6}=2\sqrt{6}$

$LHS=RHS\because$ Vectors are collinear.