$4.[$points $40]$ Given the initial word embedding from the question $3.$

"watching": $[0\mathrm{.}3,0\mathrm{.}1,0\mathrm{.}2][0\mathrm{.}3,0\mathrm{.}1,0\mathrm{.}2]$

"movies": $[0\mathrm{.}2,0\mathrm{.}4,0\mathrm{.}6][0\mathrm{.}2,0\mathrm{.}4,0\mathrm{.}6]$

"sunny": $[0\mathrm{.}7,0\mathrm{.}1,0\mathrm{.}4][0\mathrm{.}7,0\mathrm{.}1,0\mathrm{.}4]$

"football": $[0\mathrm{.}5,0\mathrm{.}3,0\mathrm{.}2][0\mathrm{.}5,0\mathrm{.}3,0\mathrm{.}2]$

"friends": $[0\mathrm{.}6,0\mathrm{.}3,0\mathrm{.}1][0\mathrm{.}6,0\mathrm{.}3,0\mathrm{.}1]$

Instructions: Follow the steps when applicable to : i$)$ Simulate the dot product between context and target vectors. ii$)$ Apply gradient descent to update the word vectors $($assume a learning rate of $0\mathrm{.}01).$ iii$)$ Perform one iteration of the embedding update.

a$)[$points $10]$ Show the word embedding updates after one iteration for the word "movies" when the context word is "watching." Show detailed computations in each step. Explain how the dot product helps capture word similarity during the training process.