Question

Find the area of the triangle with the given vertices. (Hint: \(\frac{1}{2}\|\mathbf{u} \times \mathbf{v}\|\) is the area of the triangle having \(u\) and \(v\) as adjacent sides.) $$ (1,2,0),(-2,1,0),(0,0,0) $$

Short answer

The area of the triangle is \( \frac{1}{2} \sqrt{13} \) square units.

Step-by-step solution

Find the two vectors

The two vectors can be found by subtracting the coordinates of the points. Using the given vertices, we get: \(\mathbf{u} = (1,2,0) - (0,0,0) = (1,2,0) \) and \(\mathbf{v} = (-2,1,0) - (0,0,0) = (-2,1,0)\).

Compute the cross product

The cross product of the two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is computed as \(\mathbf{u} \times \mathbf{v} = (2*0 - 0*1, 0*-2 - 1*0, 1*1 - 2*-2) = (0, 2, 3)\)

Calculate the magnitude

The magnitude of the cross product is calculated as \(\|\mathbf{u} \times \mathbf{v}\|= \sqrt{(0)^2 + (2)^2 + (3)^2} = \sqrt{13}\)

Compute the area

The area of the triangle with \(\mathbf{u}\) and \(\mathbf{v}\) as adjacent sides is calculated as \(\frac{1}{2} \|\mathbf{u} \times \mathbf{v}\|\= \frac{1}{2} \sqrt{13}\).