Find the area of a triangle with vertices: (2,0,,3,0),
and (1,1,4) using the dot product only. verify youranswer is
correct by using heron's formula.

a. The area of the triangle is

b.  Using Heron's formula the area is

Step-by-step explanation:

a. The magnitude of the product u × v is by definition the area of the parallelogram spanned by u and v when placed tail-to-tail. Hence we can use the vector product to compute the area of a triangle formed by three points A, B and C in space.

It follows that the area of the triangle is

Let A = (2, 0, 0), B = (0, 3, 0), and C = (1, 1, 4).

We have and

The cross product of two vectors and is given by

Applying the above formula we get

The magnitude of a vector is given by

Thus,

The area of the triangle is

b. The Heron's formula is given by

where and a, b, and c are the sides of the triangle.

To find the sides of the triangle given the vertices you must,

f(x) has value 150 at x= 0

and 0 at x= 3

so d(f(x))/dt = slope = -150/3 = -50