Q. 10 - Calculus

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?

Step-by-Step Solution

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 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 ℝ 3

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.

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