PIL import Image import pytesseract

Load the image

image_path = '/mnt/data/1733198707892jsfmt8e5.jpg' image = Image.open(image_path)

Use Tesseract to extract text

extracted_text = pytesseract.image_to_string(image) extracted_textThe problem requires calculating the following vector operations for each pair of vectors $u$ and $v$:

(a) $u + v$,
(b) $u - v$,
(c) $2u - 3v$.

The given vectors are:

1. $u = (2, 1), v = (1, 3)$,
2. $u = (-5, 3), v = (0, 0)$,
3. $u = i + j, v = 2i - 3j$,
4. $u = 2i, v = j$.

Let me solve these step-by-step and fill in the table. pandas as pd

Define the vectors and compute the operations

data = [ {"u": (2, 1), "v": (1, 3)}, {"u": (-5, 3), "v": (0, 0)}, {"u": (1, 1), "v": (2, -3)}, # Representing i + j as (1, 1) and 2i - 3j as (2, -3) {"u": (2, 0), "v": (0, 1)}, # Representing 2i as (2, 0) and j as (0, 1) ]

Calculate results

results = [] for entry in data: u = entry["u"] v = entry["v"] u_plus_v = (u[0] + v[0], u[1] + v[1]) u_minus_v = (u[0] - v[0], u[1] - v[1]) two_u_minus_three_v = (2 * u[0] - 3 * v[0], 2 * u[1] - 3 * v[1]) results.append({ "u + v": u_plus_v, "u - v": u_minus_v, "2u - 3v": two_u_minus_three_v, })

Create DataFrame to display results

df = pd.DataFrame(results, index=["7", "8", "9", "10"]) import ace_tools as tools; tools.display_dataframe_to_user(name="Vector Operations Table", dataframe=df)I have calculated and presented the results of the vector operations in a table for your review. Let me know if you want further details or explanations.

Here are five questions to expand your understanding:

1. What do the operations $u + v$, $u - v$, and $2u - 3v$ represent geometrically?
2. How can you generalize the calculations for vectors in higher dimensions?
3. Why is the multiplication of vectors by scalars important in vector algebra?
4. Can you verify these calculations using a graph or diagram to visualize the vectors?
5. How do these operations apply in physics or engineering contexts?

Tip: When working with complex vector problems, breaking the components into individual operations often simplifies the calculations.