The position vectors of the vertices of triangle are $ 3 \hat i + 4 \hat j + 5 \hat k $, $ \hat i + 7 \hat k $ and $ 5 \hat i + 5 \hat j $. The distance between the circumcentre and the orthocenter is?
I found the orthocenter using triangle properties and formula. But how do I find orthocenter and circumcenter using vectors. Can I find it using midpoint theorem of two vectors, and then scalar triple product?
$$\vec{a}=(3,4,5);\vec{b}=(1,0,7);\vec{c}=(5,5,0)$$ sides of the triangle are $$x=||\vec{a}-\vec{b}||=2\sqrt{6}\\y=||\vec{a}-\vec{c}||=\sqrt{30}\\ z=||\vec{c}-\vec{b}||=3\sqrt{10}$$ Half perimeter is $$p=\frac{x+y+z}{2}$$ and area $$S=\sqrt{p(p-a)(p-b)(p-c)}$$ Circumscribed circle radius is $$R=\frac{xyz}{4S}$$ distance between circumcenter and orthocenter is $$d=\sqrt{9 R^2-\left(x^2+y^2+z^2\right)}=3 \sqrt{\frac{274}{11}}$$