Question

A triangle has vertices at A(2,3,1) , B(-1,1,2) , C(1,-2,3).Find the acute angle which the median to side AC makes with side BC

Answer 1

$$\bold{r_M}=0.5(\bold{r_A+r_C})\\=0.5((2,3,1)+(1,-2,3))=(1.5,0.5,2)$$

$$\cos{\alpha}=\frac{BC\cdot BM}{|BC||BM|}\\\cos{\alpha}=\frac{(1+1,-2-1,3-2)\cdot (1.5+1,0.5-1,2-2)}{|(1+1,-2-1,3-2)||(1.5+1,0.5-1,2-2)|} \\\cos{\alpha}=\frac{(2,-3,1)\cdot (2.5,-0.5,0)}{2.55\sqrt{14}} \\\cos{\alpha}=\frac{6.5}{9.54}\\\alpha=47\degree$$