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. We're told that these points ar

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

Key Concepts

Magnitude of a Vector

The magnitude, or norm, of a vector is a measure of its length computed as the square root of the sum of the squares of its components. In the context of the cross product, the magnitude of the resulting vector gives the area of the parallelogram spanned by the two original vectors.

Area of a Triangle Using Vectors

The area of a triangle formed by two vectors can be found by taking half the magnitude of their cross product. This method leverages the geometric interpretation of the cross product, linking vector operations directly to spatial area calculations.

Vector Subtraction

Vector subtraction is the process by which one vector is subtracted from another, often used to represent the change or difference between two points in space. When finding the area of a triangle, subtracting the coordinates of one vertex from the other two gives you the vectors representing two sides of the triangle.

Cross Product

The cross product of two vectors in three-dimensional space is a vector that is perpendicular to the plane containing the original vectors. Its magnitude equals the area of the parallelogram formed by the two vectors, making it a useful tool for calculating areas in vector geometry.

Transcript

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

00:03 So we're given three points here.

00:05 We're told that these points are vertices of a triangle.

00:09 We're asked to find the area of this triangle.

00:11 So the problem gives us that the area of a triangle is equal to one half times the length of the vector that results from u cross v.

00:23 So for this problem, we need to find what the u and v vectors are.

00:28 So essentially what they are is there are two vectors that.

00:31 That represent two sides of the triangle that are adjacent to each other.

00:36 So for example, just draw an example of just a generic triangle.

00:40 If you have a triangle like that, we need to choose a starting point for a triangle.

00:45 So let's say that's our starting point.

00:48 Then we need two vectors that are adjacent that start from the same starting point.

00:54 And then one of these would represent you on one of these would represent v.

00:57 And then we take the cross product and find the area.

00:59 So in order to get these u and v vectors, we would need to use these points that are given.

01:07 So for a vector between two points, you take, so let's say, for example, this is our starting point, and we're trying to draw a vector from there to this point.

01:20 So we take the, from the second point, we take the individual vector components and subtract the corresponding vector components from the original point to get the x value of our vector.

01:35 And you do the same from the y.

01:37 You do this value minus this value for the y and this value minus this value for the z.

01:41 And you'll see more as i, um, we work out this problem.

01:44 So that's how you would get the vectors and then you find what you and v are and solve for area.

01:50 So we're going to go and dive into this problem then.

01:54 So let's use this two three minus five point is our starting point right here.

02:00 So we want to try and find, let's let the u vector represent the vector that goes from that point to this point, minus two, minus two, zero.

02:11 So for our x component of this vector, we're going to take the x point of the second point and subtract the x value of the original point.

02:21 So we've got minus two, minus two, it gives us minus four.

02:25 We're going to do the same from the y value.

02:26 We're going to take minus two and subtract 3 from that, and that gives us minus 5.

02:33 And then lastly, we're going to take 0 and subtract negative 5 from that, which gives us 5.

02:39 So this is our u vector.

02:41 So now we're going to find our v vector.

02:43 So we're going to use the same starting point, this 2, 3, negative 5, or we're going to then use this point as our second point that we're going to.

02:52 That's the, we'll form an adjacent vector.

02:55 So we're going to take, for our x component, we're going to take the x value of our second point 3 and subtract the x value of our first point two.

03:02 So we've got 3 minus 2, and that gives us 1.

03:05 Leaving on to our y component, we have 0 minus 3 gives us minus 3.

03:11 And then for a z component, we have our 6, then minus negative 5, that gives us 11.

03:23 Okay, so we have our u and v vectors.

03:25 Now we need to find u cross v.

03:28 So to do that, we're going to set up our 3x3 matrix, our first 1 ,000, our first, the first row is going to have our i, j, and k unit vectors.

03:36 Our second row is going to consist of the components of our u vector, which are minus 4, minus 5, and 5...