What is the area of the triangle which has two sides defined by the vectors

u=(2,2,1)

v=(5,2,1)

Enter your answer to two decimal places, rounding as appropriate.

Question

What is the area of the triangle which has two sides defined by the vectors

u=(2,2,1)

v=(5,2,1)

Enter your answer to two decimal places, rounding as appropriate.

BlackTom AI · Accepted Answer

Area = (1/2) |u × v| because the magnitude of the cross product of two vectors gives the area of the parallelogram they span; half of that is the area of the triangle formed by the two side vectors.

1) Relevant concepts
- Cross product in R^3: u × v is a vector perpendicular to both u and v with magnitude |u||v|sinθ equal to the area of the parallelogram spanned by u and v.
- Area of the triangle with sides u and v is half the parallelogram area: Area = 1/2 |u × v|.

2) Step-by-step computation
Given u = (2,2,1) and v = (5,2,1), compute the cross product components using
u × v = (u2 v3 − u3 v2, u3 v1 − u1 v3, u1 v2 − u2 v1).

3) Intermediate arithmetic
- First component: u2*v3 − u3*v2 = 2*1 − 1*2 = 2 − 2 = 0
- Second component: u3*v1 − u1*v3 = 1*5 − 2*1 = 5 − 2 = 3
- Third component: u1*v2 − u2*v1 = 2*2 − 2*5 = 4 − 10 = −6

So u × v = (0, 3, −6).

Magnitude: |u × v| = sqrt(0^2 + 3^2 + (−6)^2) = sqrt(9 + 36) = sqrt(45) = 3√5 ≈ 6.708203932.

Triangle area = (1/2) |u × v| = 0.5 * 6.708203932 = 3.354101966.

4) Rounding and justification
Rounded to two decimal places (standard rounding), 3.354101966 → 3.35. This value is the exact numeric result of the geometric formula.

5) Other options
Not applicable here (this is a short-answer numeric question), but any alternative numeric value would be incorrect because it would either miscompute the cross product, the magnitude, or neglect the factor of 1/2 for a triangle.

6) Conclusion
The computed and rounded area is 3.35, which matches the provided correct answer.

What is the area of the triangle which has two sides defined by the vectors u=(2,2,1) v=(5,2,1) Enter your answer to two decimal places, rounding as appropriate.简答题

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