Proof Consider the vectors 𝐮=⟨cosα, sinα, 0⟩and 𝐯=⟨cosβ, sinβ, 0⟩, where α>β. Find the cross pro

step 1 Hello, I hope you're doing well. So we're given these two vectors, U and V, and we need to, usin

Proof Consider the vectors 𝐮=⟨cosα, sinα, 0⟩and 𝐯=⟨cosβ,...

Key Concepts

Sine Difference Identity

The sine difference identity, expressed as sin(? - ?) = sin ? cos ? - cos ? sin ?, is a fundamental trigonometric identity that rewrites the sine of a difference of two angles in terms of the sines and cosines of those angles. This identity is widely used to simplify and prove trigonometric expressions and equations.

Vector Representation in the Plane

Vectors represented using trigonometric functions, where the components are given by cosine and sine of an angle, describe points on the unit circle in the plane. This form of representation makes it easier to relate geometric interpretations of rotations and angles with algebraic operations like the cross product.

Cross Product

The cross product is an operation on two vectors in three-dimensional space that results in a third vector which is perpendicular to the plane containing the original vectors. Its magnitude is given by the product of the magnitudes of the initial vectors and the sine of the angle between them, with the direction determined by the right-hand rule.

Transcript

00:02 Hello, i hope you're doing well.

00:04 So we're given these two vectors, u and v, and we need to, using this information, so our u vector, v vector, and alpha is greater than v, we need to prove that sine of the angle alpha minus beta is equal to sine alpha cosine beta minus cosine alpha sine beta.

00:26 Okay, so in order to prove this, the first thing, we just need to review a few things beforehand.

00:32 So first thing we need to review is how to find the cross product of two vectors.

00:37 So say that v cross u, so you're finding that cross product, to find the cross product, we're going to set up a three by three matrix and find its determinant.

00:46 So the first row is going to be the unit vectors, i, j, and k.

00:49 Second row is going to be the components of their first vector, which is v.

00:54 So v1, v2, b3.

00:56 The third row will be the components of our second vector.

00:59 In this case, it's u, so u1, u2, u3.

01:02 Then to find the cross product, you're going to take the i unit vector and multiply by the determinant of this 2x2 matrix, then subtract the j vector multiplied by the determinant of this 2x2 matrix, and add the k vector multiplied by the determinant of this 2 by 2 matrix.

01:20 Then you'll simplify that and end up with the cross product.

01:24 Okay, so the other thing we need to review is how to find the length of vectors.

01:31 The length of any vector is going to be the square root of the sum of the squares of the vectors components.

01:36 So in this case, it would be u1 squared plus u2 squared plus u3 squared.

01:43 We also need to remember that the value of any cross product is equal to, so we want to get the value of u cross v.

01:58 So the length of the vector that results from the cross product is equal to, the length of u multiplied by the length of v multiplied by sine theta.

02:10 So whatever that, whatever your angle is between the two vectors.

02:17 Lastly, we need to remember that sine squared of theta plus cosine squared of theta is equal to 1.

02:24 Okay, so keeping all that in mind, let's go and take our cross product v cross u and see what we get.

02:30 So v -u, we're going to set up our 3x3 matrix.

02:38 We've got i, j, k, and the first row.

02:42 Second row, we've got our components for a v vector, which is cosine, beta, sine, beta, and 0.

02:47 So cosine, data, sine beta, and zero.

02:52 And for our third row, we're going to write our components of our u vector.

02:55 That's cosine alpha, sine alpha, and zero.

02:57 So cosine alpha, sine alpha, and zero.

03:02 And then we're going to take the determinant of this.

03:06 So before we get started, i just want to make sure you understand that, or let's just draw out our vectors just to get a good visual of what we're working.

03:15 So we know that we have two vectors.

03:22 Because we have components, cosine alpha, sine alpha, we've got cosine beta, sine data.

03:31 Since sine squared theta plus cosine squared theta is equal to 1, when you find the value the length of the vector u, squared a u1 squared plus u2 squared plus u3 squared.

03:43 That will be squared of cosine squared alpha plus sine squared alpha, which will give us once.

03:48 That means value of u is equal to one.

03:51 And same with v.

03:53 We'll have square to cosine squared beta plus squared, i'm sorry, square to cosine squared beta plus sine squared beta, according to this identity, b is equal to 1 as well.

04:02 So these are two vectors with units of length of 1.

04:05 So we've got length of 1, length of 1.

04:08 And so this means that this x component is cosine of the angle that it forms, y component is sign of the angle that it forms.

04:16 This just means that the angle with the horizontal axis of our, we know that alpha is greater than beta.

04:24 So this first angle here is going to be beta.

04:28 And this second angle there is going to be alpha.

04:33 So that means we have our u is our angle that deals with the alpha, has the angle alpha, or v is the angle that has, or vector that has the angle beta.

04:44 So this is what u and v look like.

04:45 So the reason we're doing v cross u and not u cross v is if you stick your fingers in the direction of you, turn them towards v, your thumb will be pointing down, whereas if you do v, that would be u cross u...

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