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Let p1 = (4,0,4), p2 = (2,-1,8), and p3 = (1,2,3).

a) Show that the three points define a right triangle. Hint the difference between two vertices is a vector whose direction coincides with that of a triangle side, and a pair of such vectors must be orthogonal in order for the triangle to be a right triangle.

b) Specify a vector N that is normal to the plane of p1, p2, and p3. Hint: N must be orthogonal to p2-p1, p3-p2, and p1-p3.

c) Specify the area of the triangle defined by p1, p2, and p3.

d) Specify the condition that p = (x,y,z) lies in the plane of p1, p2, and p3 (as an equation in x, y, and z). Recall that the equation of a plane has the form Ax + By + Cz + D = 0.

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