# Cosine: products, lengths and division

Subject: Natural Language Processing

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/nlp/#tee/formulas/f-vectors/u2-cosine

Canonical: https://deep-learning-study-room.vercel.app/read/nlp/tee/formulas/f-vectors--u2-cosine/

Formulas explained through full worked examples

**Cosine similarity compares the direction of two nonzero vectors.** Multiply matching coordinates and add to obtain the dot product. To obtain each vector's norm, square its coordinates, add them and take a square root.



$$
\cos(x,y)=\frac{\sum_i x_i y_i}{\sqrt{\sum_i x_i^2}\sqrt{\sum_i y_i^2}}.
$$



The vectors must use the same coordinate/vocabulary order. For nonnegative BoW or TF–IDF vectors, the result lies from 0 to 1; for general real-valued vectors, it can range from −1 to 1. A zero vector makes this formula undefined.

**Question. Find cosine similarity for x=[1,1] and y=[1,0].**

#### Given

Nonzero vectors, matching coordinate order.

#### To find

Find cosine similarity for x=[1,1] and y=[1,0].

#### Formula and why it applies

Use 

$$
\cos(x,y)=(x\cdot y)/(\|x\|\|y\|)
$$

 because direction similarity is required.

#### Step 1. Multiply matching coordinates

1 × 1 = 1; 1 × 0 = 0.

#### Step 2. Add products

1 + 0 = 1.

#### Step 3. Length of x



$$
\sqrt{(1)^2+(1)^2}=\sqrt{2}.
$$



#### Step 4. Length of y



$$
\sqrt{(1)^2+(0)^2}=\sqrt{1}.
$$



#### Step 5. Divide



$$
\frac{1}{\sqrt{2}\sqrt{1}}\approx 0.707107.
$$



**Answer.** Cosine≈0.707107.

**Question. Find cosine similarity for x=[1,2] and y=[2,4].**

#### Given

Nonzero vectors, matching coordinate order.

#### To find

Find cosine similarity for x=[1,2] and y=[2,4].

#### Formula and why it applies

Use 

$$
\cos(x,y)=(x\cdot y)/(\|x\|\|y\|)
$$

 because direction similarity is required.

#### Step 1. Multiply matching coordinates

1 × 2 = 2; 2 × 4 = 8.

#### Step 2. Add products

2 + 8 = 10.

#### Step 3. Length of x



$$
\sqrt{(1)^2+(2)^2}=\sqrt{5}.
$$



#### Step 4. Length of y



$$
\sqrt{(2)^2+(4)^2}=\sqrt{20}.
$$



#### Step 5. Divide



$$
\frac{10}{\sqrt{5}\sqrt{20}}\approx 1.000000.
$$



**Answer.** Cosine≈1.000000.

**Question. Find cosine similarity for x=[1,2,0] and y=[0,1,2].**

#### Given

Nonzero vectors, matching coordinate order.

#### To find

Find cosine similarity for x=[1,2,0] and y=[0,1,2].

#### Formula and why it applies

Use 

$$
\cos(x,y)=(x\cdot y)/(\|x\|\|y\|)
$$

 because direction similarity is required.

#### Step 1. Multiply matching coordinates

1 × 0 = 0; 2 × 1 = 2; 0 × 2 = 0.

#### Step 2. Add products

0 + 2 + 0 = 2.

#### Step 3. Length of x



$$
\sqrt{(1)^2+(2)^2+(0)^2}=\sqrt{5}.
$$



#### Step 4. Length of y



$$
\sqrt{(0)^2+(1)^2+(2)^2}=\sqrt{5}.
$$



#### Step 5. Divide



$$
\frac{2}{\sqrt{5}\sqrt{5}}\approx 0.400000.
$$



**Answer.** Cosine≈0.400000.

Proportional vectors [1,2] and [2,4] have cosine 1 but nonzero Euclidean distance. Direction similarity does not require identical magnitudes.

Cosine: matching products add upar; dono lengths multiply neeche.