Hints

Use these only after trying the exercises.

MATH-C02-C-001

Read the vector entries in order.

Addition uses matching coordinates.

The scalar multiplies every coordinate.

Dot product means multiply matching entries, then add.

Distance between two positions is the length of their difference.

For $[1,0]$ and $[0,1]$ , the dot product is 0.

Projection onto $[1,0]$ keeps the horizontal part.

The dot product collapses matching products into one number.

Embeddings are learned representations, not guaranteed coordinate dictionaries.

Cosine similarity compares direction after dividing by vector lengths.

Ask whether the first coordinate means the same thing in both vectors.

Residual means original vector minus projected vector.