Question

What is meant by the triangle determined by vectors u and v in ${\mathrm{ℝ}}^3$? How do you find the area of this triangle?

Short answer

Vectors u and v are visualized as directed line segments whose lengths are their magnitudes.

$\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{triangle}\mathrm{determined}\mathrm{by}\mathrm{the}\mathrm{two}\mathrm{vectors}=\frac{1}{2}\left|\overrightarrow{A}\times \overrightarrow{B}\right|$

Step-by-step solution

Step 1. Given Information 

What is meant by the triangle determined by vectors u and v in ${\mathrm{ℝ}}^3$? How do you find the area of this triangle?

Step 2. Firstly explaining meant by the triangle determined by vectors u and v in ℝ3

Vectors u and v are visualized as directed line segments whose lengths are their magnitudes. We will use this concept well in this concept explanation, the area of a triangle formed by vectors. Normally when we try to find out the area of a triangle, we usually find out the value by the formula of Heron’s Formula.

Step 3. We can express the area of a triangle by vectors also. 

$\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{triangle}\mathrm{determined}\mathrm{by}\mathrm{the}\mathrm{two}\mathrm{vectors}=\frac{1}{2}\left|\overrightarrow{A}\times \overrightarrow{B}\right|$