# Q4A · Bag of Words and three cosine similarities

Subject: Natural Language Processing

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/nlp/#tee/mocks/timed-reexam-2026/reexam-2026-q4a

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TEE · actual previous paper · 100 marks · 180 minutes

Question. Consider the following three documents:  
Document A: “The dog chased the cat.”  
Document B: “The cat chased the mouse.”  
Document C: “The dog ran after the cat.”  
Using the Bag of Words (BoW) model, represent each document as a vector after removing the stop words {The, by, was}.  
i. Construct the term-document matrix.  
ii. Compute the cosine similarity between each pair of documents (A–B, A–C, and B–C).  
iii. Interpret and comment on the results obtained in terms of document similarity. [10 marks]

### Answer

### Given

Three documents. Lowercase and remove punctuation. Remove the specified stop words case-insensitively; “the” is removed everywhere. Keep “after” because it is not in the supplied stop list. Use raw word counts.

### To find

A common-vocabulary count matrix, three cosine values and their interpretation.

### Formula: which one and why



$$
\mathrm{BoW}_j(d)=C(v_j,d)
$$



vⱼ is vocabulary term j. The coordinate stores its count in document d; order in the sentence is not retained.



$$
\cos(\mathbf a,\mathbf b)=\frac{\sum_j a_jb_j}{\sqrt{\sum_j a_j^2}\sqrt{\sum_j b_j^2}}
$$



### Solution

### Step 1. Clean each document

A: dog chased cat.  
B: cat chased mouse.  
C: dog ran after cat.

### Step 2. Fix vocabulary and matrix

Vocabulary: dog, chased, cat, mouse, ran, after.

| Term | A | B | C |
| --- | --- | --- | --- |
| dog | 1 | 0 | 1 |
| chased | 1 | 1 | 0 |
| cat | 1 | 1 | 1 |
| mouse | 0 | 1 | 0 |
| ran | 0 | 0 | 1 |
| after | 0 | 0 | 1 |

A = [1,1,1,0,0,0]; B = [0,1,1,1,0,0]; C = [1,0,1,0,1,1].

### Step 3. Calculate vector lengths



$$
\|A\|=\sqrt{1^2+1^2+1^2}=\sqrt3
$$





$$
\|B\|=\sqrt{1^2+1^2+1^2}=\sqrt3
$$





$$
\|C\|=\sqrt{1^2+1^2+1^2+1^2}=2
$$



### Step 4. Calculate A–B



$$
A\cdot B=(1)(0)+(1)(1)+(1)(1)+(0)(1)+(0)(0)+(0)(0)=2
$$





$$
\cos(A,B)=\frac2{\sqrt3\sqrt3}=\frac23\approx0.666667
$$



### Step 5. Calculate A–C



$$
A\cdot C=(1)(1)+(1)(0)+(1)(1)+(0)(0)+(0)(1)+(0)(1)=2
$$





$$
\cos(A,C)=\frac2{\sqrt3\cdot2}=\frac1{\sqrt3}\approx0.577350
$$



### Step 6. Calculate B–C



$$
B\cdot C=(0)(1)+(1)(0)+(1)(1)+(1)(0)+(0)(1)+(0)(1)=1
$$





$$
\cos(B,C)=\frac1{\sqrt3\cdot2}\approx0.288675
$$



### Final answer

**A–B = 0.666667; A–C = 0.577350; B–C = 0.288675.** A and B have the highest lexical overlap under this representation. B and C share only cat. BoW similarity reflects token overlap, not full semantic equivalence or word order.

**Yaad rakho:** Same vocabulary order rakho. Corresponding counts multiply-add; phir dono lengths se divide.