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 the vertices of a tria

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

Key Concepts

Cross Product

The cross product of two vectors in three-dimensional space is a vector that is perpendicular to both original vectors. Its magnitude is equal to the area of the parallelogram formed by the vectors. This operation is fundamental when determining the area of a triangle since the area can be derived by taking half the magnitude of this cross product.

Area of a Triangle Using the Cross Product

This key concept uses the property that the area of a triangle is half the area of the parallelogram defined by two adjacent side vectors. It is expressed mathematically as A = 1/2 ||u × v||, where u and v represent vectors along the sides of the triangle. This formula is particularly useful in problems involving coordinates in three-dimensional space.

Vector Subtraction

This concept involves finding the vector that represents the direction and magnitude between two points by subtracting the coordinates of one point from the other. In the context of triangles in space, it is used to form side vectors from given vertices, which are essential for further calculations like the cross product.

Magnitude of a Vector

This concept refers to the length or norm of a vector, which is calculated using the square root of the sum of the squares of its components. In the context of finding areas, the magnitude of the cross product gives the area of the parallelogram, and it is necessary for converting this into the triangle's area.

Transcript

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

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

00:07 And we need to find the area of the triangle.

00:09 So the problem tells us that the area of the triangle is equal to one half times the length of the cross product of u crossb.

00:20 So we need to find out what u and v are for this triangle.

00:25 So the u and v vectors are essentially two vectors that come from the same point in a triangle.

00:30 And so they're adjacent vectors in the triangle.

00:34 So, for example, let's just draw like a generic triangle here.

00:42 So let's choose this right here is our starting point.

00:46 So then we can draw a vector to the next point, and that's vector u.

00:51 And then we draw a vector to the other point that it connects to, and that's a vector v.

00:54 Then we can take the cross product and find the area.

00:57 So we need to figure out what u and v are this triangle.

01:02 So generally how you would find you and be is you would take the point here, so it would have x1, y1, y1, z1.

01:14 You would take the point here, x2, y2, z2, and to find the vector between this point 1 and point 2, your vector, your first component of your vector would be x2 minus x1, second component would be y2 minus y1, and third would be z2 minus z1.

01:30 So that's how you find the vector that connects the two points.

01:34 And then you do the same thing to find v.

01:36 Then you plug that in here.

01:38 So let's go in, you apply that to this problem.

01:41 So first we need to figure out what our starting point is.

01:44 So let's use this equation or this point as our starting point, just because it's the lowest, so the math will work out a little easier.

01:52 So it will end up with more positive numbers.

01:55 So first going to find our u vector.

01:58 So let's make u vector the point that goes from the vector that goes from this point to this point up here.

02:04 So what we're going to do is we're going to take, for our x component, we're going to take the x value of this point and then subtract the x value of, i'm sorry, not that.

02:16 You're going to take the x value of the second point, and from that you're going to subtract the x value of the first point.

02:22 So you've got two minus negative two gives us four.

02:25 For a y point, we do four minus negative four, gives us eight, for our next point.

02:31 We do 0 minus 0 gives us 0.

02:34 So that's our uvector.

02:36 So now we're going to do our v vector.

02:38 So this time it's going to start at the same starting point, but we're going to go to this point instead is our second point.

02:45 So for our x component, we're going to take the x value of our second point and subtract the x value of our first point.

02:50 So we've got 0 minus negative 2 gives us 2.

02:53 We have 0 minus negative 4 gives us 4.

02:56 And we've got 4 minus 0 gives us 4 as well.

03:00 So we have our u and v vectors.

03:03 So now we need to figure out what u cross v is.

03:06 So to do that, we're going to create a 3x3 matrix, that i, j, u, k.

03:14 Then for our first row, then for our second row, we're going to write our components of our u vectors, so it's 480...

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