Consider the points below. P(2, 0, 2),  Q(−2, 1, 3),  R(4, 2, 4)
(a) Find a nonzero vector orthogonal to the plane through the points
P, Q,
and R.

a) The nonzero vector orthogonal to the plane through the points P, Q and R is (0, 10, -10).

b) The area of the triangle PQR is .

Step-by-step explanation:

The statement is incomplete. The complete question will be presented below:

Consider the points below: P(2, 0, 2),  Q(−2, 1, 3),  R(4, 2, 4)

(a) Find a nonzero vector orthogonal to the plane through the points

P, Q,  and R. (b) Find the area of the triangle PQR.

a) Let , and . First, we construct vectors PQ and PR by using the following formulas:

(1)

(2)

Then, we can find the nonzero vector orthogonal to the plane by means of the following cross product:

Which can be solved by the following determinant (Law of Sarrus):

The nonzero vector orthogonal to the plane through the points P, Q and R is (0, 10, -10).

b) The area of the triangle is determined by the following formula:

(3)

The area of the triangle PQR is .

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