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#  Visualizing Vector Arithmetic Lab

## ![Spark Logo Tiny](https://files.training.databricks.com/images/105/logo_spark_tiny.png) In this lab you:<br>

* Apply and visualize basic vector arithmetic to embeddings

* Calculate cosine similarity between vectors

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We need the gensim library to load in the pretrained GloVe vectors again.

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We are going to recreate the DataFrame and graph showcasing the embeddings of the words "man", "woman", "king", and "queen."

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Now we want to be able to visualize the vector arithmetic involved with the `word_vectors.most_similar(positive=['woman', 'king'], negative=['man'])` call. The call will return the word with the embedding which is closest to the vector resulting from adding the embeddings in `positive` and subtracting the embeddings in `negative`. 

To visualize what the resulting vector should be, we will continue to work with the reduced 2D representations of the pretrained GloVe embeddings and first add the positive embeddings `woman` and `king` before subtracting the embedding of `man`. 

Fill in the following 2 lines to get 

1. the intermediate `woman + king` embedding 

2. the final `woman + king - man` embedding

and run the cell to plot the original 4 vectors with the 2 newly calculated vectors.

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Even though we reduced the embeddings down to 2 dimensions, what word does the resulting embedding still look the closest to? Does this match the answer that `word_vectors.most_similar(positive=['woman', 'king'], negative=['man'])` returns?

Note: Since vector addition and subtraction are commutative, you can change the order of the operations and see that the resulting `woman+king-man` vector still approximates the `queen` embedding.

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Now we are going to explore what exactly "similar" embeddings mean.

Recall the definition of cosine similarity from above:

$$\text{CosineSimilarity }(u,v) = \cos(\text{ angle between vectors }) = \frac{u \cdot v}{|u| |v|}$$

Run the following cell to see what the `gensim` default function returns for the similarity between `queen` and `king`.

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Now fill in the following `cos_similarity` function to calculate the cosine similarity between 2 vectors (np arrays) of equal dimensions.  

Hint: Take a look at some numpy functions.

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Let's use the similarity measure that we implemented to be able to pick a word, out of an inputted list, that is "closest" to a given word.

Fill out the below function `most_similar` to return the word from `words` that has the closest cosine similarity to word `w`.

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You can play around with the inputs to `most_similar` and see which words the GloVe embedding model treats has having similar semantic meanings.