In Exercises 43-46, find the area of the triangle with the given vertices. The area A of the tri

step 1 Hello, hope you're doing well. So we're given three points here that form a triangle. So what we

In Exercises 43-46, find the area of the triangle with the...

Key Concepts

Vectors in 3D Space

Vectors in three-dimensional space are ordered triplets that represent both a magnitude and direction. They are essential in describing positions, displacements, and directions in geometry and physics, allowing us to work with spatial relationships precisely.

Vector Subtraction

Vector subtraction is the operation used to find the difference between two vectors. In the context of geometry, subtracting the coordinates of one point from another yields a new vector, which represents the direction and distance from the first point to the second, typically used to form sides of geometric figures like triangles.

Cross Product

The cross product is an operation on two vectors in three-dimensional space that results in another vector perpendicular to both initial vectors. Its magnitude equals the area of the parallelogram that the vectors span, making it a powerful tool in computing areas of shapes in space.

Triangle Area via Cross Product

Using the cross product to compute the area of a triangle involves forming two side vectors from the vertices, taking their cross product, and then halving the magnitude of the resulting vector. This method leverages the geometric interpretation of the cross product to determine the area efficiently.

Transcript

00:01 Hello, hope you're doing well.

00:03 So we're given three points here that form a triangle.

00:07 So what we're asked to do is we're asked to find the area of this triangle with these points.

00:12 So the way we're going to do that is we're given by this problem that the formula for an area of a triangle is one -half times the length of the vector that's the cross -product of u -cross -v.

00:24 So this u -cross -v here, and u and v are any vectors that are adjacent on this triangle.

00:30 So we need to find these vectors.

00:33 So what we're going to do is we're going to start with from one point in the triangle.

00:37 So say we have a triangle like this.

00:41 I'm going to start with one point in the triangle here.

00:44 We're going to form two vectors connecting to the other two points.

00:48 This will be our u here and this will be our v here.

00:54 And then we'll, so we'll find these vectors by, let's say we're taking these two points and finding the vectors between them.

01:02 So we're going to take, we'll start with the x value of the vector, and we're going to take the x value of the second point and subtract the x value of the first point from it.

01:15 So if the vectors from this point to this point, then it's the values in the second point minus values in the first point.

01:21 And that'll give us our vector.

01:22 So once we get our u vector and our v vectors, so our two vectors starting from the same point, we find u cross v, then we find the length of this vector, divided by two, and that gives us our area.

01:40 So that's what we're going to be doing.

01:42 All right, so now let's dive into our problem.

01:44 So we're given these three points.

01:47 We need to find two vectors that come from a common point.

01:50 So let's choose our vector for our point zero zero is kind of our starting point, because that's pretty easy vector to work with.

01:59 So our u vector is going to be the vector from 0 -00 to 123.

02:07 Let's go with 1 -2 -3.

02:09 So the way we find the points for that for this vector is for the x -value, we're going to take the x -value of this vector and subtract the x -value of this vector.

02:19 So we've got one, i'm sorry, at this point and subtract the x -value of this point.

02:24 So we have 1 minus 0 is 1.

02:26 For the next point, we have 2 minus 0 is 2.

02:31 For the next point, we have 3 minus 0 is 3.

02:35 So that is our u vector.

02:37 So now we need to find, i'm sorry, now we need to find our v vector.

02:42 So we're going to do the same thing.

02:43 We're going to start from this 0 -0.

02:45 Now we're going to find the vector between that point in this minus 3 -0.

02:51 So that's our other point in the triangle.

02:54 So to do that, we're going to take each value in this point and subtract the corresponding value from our starting point.

03:02 So for our x value, we have minus 3 minus 0.