# Thread: Area of triangle using vectors

1. ## Area of triangle using vectors

I have to find the volume of the triangle with vertices (0,0,0) (1,1,1) and (0,-2,3)

Do i define two vectors then find a normal vector. Then take 1/2 |determinant| of the 3 vectors (the two I defined and the normal) to find the volume?

2. ## Re: Area of triangle using vectors

Volume of a two-dimensional figure is called area.

You could do the thing you described, but you need to normalize the normal vector, i.e., divide it by its length. But the length of the vector product already equals twice the triangle area.

3. ## Re: Area of triangle using vectors

you are right, the set of questions ask for volume or area, this question states triangle so I take that as an area problem not volume.

so if I have a 2x2 matrix ( two vectors in an x,y plane) the the absolute value of the determinant is the area of the parallelogram and half that would be the area of a triangle formed by cutting the parallelogram in half

but in this instance I define two vectors in 3-space. so take the cross product and times it by 1/2 and thats the area. correct?

$\displaystyle <1,1,1> X <0,-2,3> \rightarrow \frac{1}{2}\sqrt{5^2+3^2+(-2)^2} = \sqrt{38}/2$

4. ## Re: Area of triangle using vectors

Yes. The second coordinate of the cross product is -3 rather than 3, but, of course, this does not change the length.