Area of a triangle Find the area of the triangle with vertices on the coordinate axes at the poi

Area of a triangle Find the area of the triangle with...

Key Concepts

Triangle Area Calculation in 3D

In three-dimensional space, the area of a triangle is found by taking one-half the magnitude of the cross product of two side vectors. This method is a generalization of the two-dimensional triangle area formula and efficiently handles the complexity of spatial geometry.

Vector Representation and Subtraction

When dealing with points in space, each vertex is represented by a position vector, and the sides of the triangle can be determined by subtracting these vectors. This operation gives the direction and magnitude of the edges, laying the foundation for further geometric computations.

Cross Product

The cross product in three dimensions results in a vector whose magnitude equals the area of the parallelogram formed by the original two vectors. It is a fundamental tool for finding areas when working with spatial vectors, as it inherently incorporates both the magnitudes and the angles between the vectors.

Transcript

00:01 Hello everybody.

00:03 In this video, i'll be showing you how to solve exercise 61 in chapter 13, section 4 of calculus early transcendentals.

00:11 Now in this problem, we are given three points a -0 -0 -0 -b -0 as well as 0 -0 -c, and we are asked to find the area of the triangle formed by these three points.

00:24 Now the first thing we want to notice is that this triangle represents half the area of a larger parallelogram that is formed by these two vectors here.

00:42 And notice that this area we've added on has the same area as our original triangle.

00:50 Now let's give a couple names to these vectors.

00:53 For this vector in blue, which extends from a -0 -0 -0 -0, let's call u.

01:00 And the vector in red that extends from a -0 -0 -0 -0 -c, it's called v.

01:08 Then we can say that u and v form this parallelogram.

01:12 And therefore a, the area of the triangle, is equal to one -half the area of the parallelogram, which is the magnitude of the cross -product of u and v.

01:28 So what we want to do is calculate this cross -product, find its magnitude, and then have it to find our area.

01:36 The first thing we want to do is identify exactly what these vectors are.

01:42 We have that u starts at a -0 and ends at 0 -b -0.

01:49 So the first component is 0 minus a, negative a, then b minus 0, which is b, and then 0 minus 0, which is 0.

02:00 And now v is equal to 0 minus a, which is negative a, 0 minus 0, which is 0, and then finally, c minus 0, which is c...

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