Answer: To find a vector perpendicular to the triangle, you can use the cross product of two vectors formed by subtracting two of the triangle's vertices. Specifically, if two vectors v1 and v2 are formed by subtracting one vertex from the other two, then their cross product v1 × v2 will be perpendicular to both v1 and v2, and therefore perpendicular to the triangle they form.
Mathematically, the cross product of two vectors in three-dimensional space is defined as:
v1 × v2 = ⟨y1z2 − z1y2, z1x2 − x1z2, x1y2 − y1x2⟩
So, to find a vector perpendicular to the triangle with vertices (x1,y1,z1), (x2,y2,z2), and (x3,y3,z3), you can choose any two vertices and compute their difference to get two vectors, and then take their cross product. For example, you could choose:
v1 = ⟨x2−x1,y2−y1,z2−z1⟩
v2 = ⟨x3−x1,y3−y1,z3−z1⟩
And then compute their cross product:
v1 × v2 = ⟨(y2−y1)(z3−z1) − (z2−z1)(y3−y1), (z2−z1)(x3−x1) − (x2−x1)(z3−z1), (x2−x1)(y3−y1) − (y2−y1)(x3−x1)⟩
This will give you a vector that is perpendicular to the triangle.
Explanation: