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, and the problem tells us that t

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

Key Concepts

Magnitude Calculation

Calculating the magnitude (or norm) of a vector involves taking the square root of the sum of the squares of its components. In the context of the triangle's area, the magnitude of the cross product of two side vectors gives a measure of the parallelogram’s area, which is then halved for the triangle.

Triangle Area Calculation

The area of a triangle formed by two vectors can be found using the formula A = 1/2 ||u x v||, where u and v are adjacent sides of the triangle. This method leverages the geometric interpretation of the cross product and effectively computes the enclosed area without the need for traditional geometric base-height measurements.

Vector Subtraction

To determine the sides of a triangle given its vertices, one must subtract the coordinates of one vertex from another. This process produces vectors that represent the directed line segments (or edges) of the triangle, which are essential for further calculations, such as finding the area using vector operations.

Cross Product

The cross product of two vectors in three-dimensional space yields a new vector that is perpendicular to both original vectors. Its magnitude equals the area of the parallelogram defined by the two vectors, making it a fundamental tool in computing areas in vector calculus.

Transcript

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

00:03 So we're given three points here, and the problem tells us that these vertices form a triangle.

00:09 So we need to try and find the area of this triangle.

00:13 So the problem tells us that the area of a triangle is given by one half times the length of the vector given by u cross v.

00:21 So u and v are two vectors that are adjacent to each other that form a triangle.

00:28 So we need to find these two vectors and what these two vectors.

00:31 Are in this triangle.

00:33 So the way we're going to do that is we're going to have, so let's say just kind of sketch out the triangle.

00:41 So we're going to choose a starting point like that.

00:44 And then we're going to find two vectors that connect that starting point with the other two points.

00:52 And this, and one will just define one of them as you and one of them as v.

00:57 So the way we're going to find that vector is we're going to take our starting point.

01:01 Let's say this is our starting point right here.

01:03 And then we're going to choose our other point that we're going to.

01:08 And for the vector that's between those two points, for each, we're going to take each component of the first point or of the second point and subtract the corresponding component of the first point from that.

01:24 And so we're going to do that with the x values, y values, and z values, and subtract those two points to form our vector.

01:30 So we're going to do that for both of these lines.

01:33 Both u and v.

01:35 Once we get them, we can take the cross product, find their value, and divide by 2, and we get the area of our triangle.

01:43 All right, so we're going to go and dive into this problem.

01:45 So let's use this as our starting point.

01:48 This 1 minus 4 .3 is our starting point, just because it's the first point.

01:53 So let's make our u vector, the vector between our starting point and this second point, 2 .2.

01:59 So to find what our vector u is, we're going to first take the x value of our second point 2, and so we'll find our 2.

02:04 Subtract the x value of our first point 1 from that.

02:08 So we've got 2 minus 1, that's 1.

02:10 Moving on to our y value, we're going to take our second point 0 minus our first point negative 4.

02:16 So 0 minus negative 4 gives us 4.

02:19 And then finally for our last, for a z component, we have 2 minus 3 gives us minus 1.

02:26 So that's our view vector.

02:27 So now moving on to our view vector, we're going to do the same thing except our second point is going to be this point, minus 2 to 0.

02:35 So our x component is going to be minus 2, minus 1, gives us minus 3.

02:41 Our y is going to be, or y component is going to be 2 minus negative 4.

02:46 It gives us 6.

02:48 And then our z component is going to be 0 minus 3, which gives us minus 3.

02:54 Okay, so now we have our u and v vectors.

02:57 So we need to find u cross v.

02:59 So to do that, we're going to set up a 3x3 matrix.

03:02 Our first row is going to consist of the unit vectors i, j, and k.

03:07 Our second row will contain the components of our u vector, which are 1, 4, negative 1...

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