Answer:
(5/2)√13 ≈ 9.01 square units
Explanation:
You want the area of ∆PQR defined by vertices P(2, 0, 2), Q(-1, -1, 0) and R(2, 3, -2).
Area
The area of the triangle will be half the magnitude of the cross product of two vectors representing the sides of the triangle:
A = 1/2|PQ×PR|
The attached calculator display shows the result of this calculation.
A = (5/2)√13 ≈ 9.01 . . . . square units
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Additional comment
The area of a triangle with side lengths 'a' and 'b' and angle C between them is A=1/2(a·b·sin(C)). The magnitude of the cross product of A and B is ...
|A×B| = |A|·|B|·sin(θ)
where θ is the angle between the vectors. Comparing these formulas, we see that the area of the triangle of interest can be found using the cross product of the vectors representing sides of the triangle. A scale factor of 1/2 is required.
If the vectors represent two adjacent sides of a parallelogram, then its area can be found using the cross product without the scale factor.
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