Furthermore, we can use the dot product definition < x, y >= |x||₂||y|l₂ cos(0) to further determine the angle between two vectors. For example, see the below image for computing and formula for computing the angle between two vectors using the dot product definition Machine learning often uses this idea of cosine similarity and the angle between vectors to compute the similarity between different data samples using their feature vectors (i.e., the columns of a dataset)! See examples here! Given the following two vectors find the angle between them (in degrees). Use the above equation, in the picture, but use the L2 norm or ||- ||₂ when computing x and y. Recall to compute the L2-norm for a vector x, we do the following: ||x|| 2 a+²+...+ 2= O 180 0-1 01 0 90 e arccos(x+y/1x1131) = y = -2
Furthermore, we can use the dot product definition < x, y >= |x||₂||y|l₂ cos(0) to further determine the angle between two vectors. For example, see the below image for computing and formula for computing the angle between two vectors using the dot product definition Machine learning often uses this idea of cosine similarity and the angle between vectors to compute the similarity between different data samples using their feature vectors (i.e., the columns of a dataset)! See examples here! Given the following two vectors find the angle between them (in degrees). Use the above equation, in the picture, but use the L2 norm or ||- ||₂ when computing x and y. Recall to compute the L2-norm for a vector x, we do the following: ||x|| 2 a+²+...+ 2= O 180 0-1 01 0 90 e arccos(x+y/1x1131) = y = -2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![Furthermore, we can use the dot product definition \( \langle x, y \rangle = \| x \|_2 \| y \|_2 \cos(\theta) \) to further determine the angle between two vectors. For example, see the below image for computing and formula for computing the angle between two vectors using the dot product definition.
### Diagram Explanation
The diagram shows two vectors, \( x \) and \( y \), with an angle \( \theta \) between them. The formula depicted is:
\[ \theta = \arccos \left( \frac{x \cdot y}{|x| |y|} \right) \]
Machine learning often uses this idea of cosine similarity and the angle between vectors to compute the similarity between different data samples using their feature vectors (i.e., the columns of a dataset!) See examples [here](#).
#### Problem
Given the following two vectors, find the **angle** between them (in degrees). Use the above equation, in the picture, but **use the L2 norm** or \(\|\cdot \|_2\) when computing \(|x|\) and \(|y|\). Recall to compute the L2-norm for a vector x, we do the following:
\[ \| x \|_2 = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2} \]
Vectors:
\[ x = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \quad y = \begin{bmatrix} -2 \\ -2 \end{bmatrix} \]
#### Options
- 180
- -1
- 1
- 90](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e96339b-d3d9-4039-844d-3fc091742cc5%2F3ac2cd44-9f5e-4494-8018-db76938d3955%2Fd2tqc9q_processed.png&w=3840&q=75)
Transcribed Image Text:Furthermore, we can use the dot product definition \( \langle x, y \rangle = \| x \|_2 \| y \|_2 \cos(\theta) \) to further determine the angle between two vectors. For example, see the below image for computing and formula for computing the angle between two vectors using the dot product definition.
### Diagram Explanation
The diagram shows two vectors, \( x \) and \( y \), with an angle \( \theta \) between them. The formula depicted is:
\[ \theta = \arccos \left( \frac{x \cdot y}{|x| |y|} \right) \]
Machine learning often uses this idea of cosine similarity and the angle between vectors to compute the similarity between different data samples using their feature vectors (i.e., the columns of a dataset!) See examples [here](#).
#### Problem
Given the following two vectors, find the **angle** between them (in degrees). Use the above equation, in the picture, but **use the L2 norm** or \(\|\cdot \|_2\) when computing \(|x|\) and \(|y|\). Recall to compute the L2-norm for a vector x, we do the following:
\[ \| x \|_2 = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2} \]
Vectors:
\[ x = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \quad y = \begin{bmatrix} -2 \\ -2 \end{bmatrix} \]
#### Options
- 180
- -1
- 1
- 90
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