Solutions

These solutions collect the exercise reasoning in one place.

MATH-C02-C-001

In:

v = [-3, 6]

the second coordinate is 6.

Add matching coordinates:

a + b = [2 + 5, -1 + 4] = [7, 3]

The first coordinate is 7.

Scale each coordinate:

3[4, -2] = [12, -6]

The first coordinate is 12.

Multiply matching entries, then add:

a \cdot b = (1 \times 2) + (3 \times 5) = 17

First compute the difference:

b - a = [4 - 1, 5 - 1] = [3, 4]

Then compute its length:

\|b-a\| = \sqrt{3^2 + 4^2} = 5

The dot product is:

[1,0]\cdot[0,1] = 0

Both vectors have length 1, so the cosine similarity is:

\frac{0}{1\times1} = 0

Projection onto $[1,0]$ keeps the horizontal component:

\text{proj}_{[1,0]}([7,-2]) = [7,0]

The dot product returns a scalar.

It multiplies matching entries, then adds the products into one number. That is why it can act as a score.

The safer statement is:

Embeddings are learned coordinates whose directions and relationships can be useful.

An embedding can be useful without every coordinate having a simple human label.

Cosine similarity compares direction after accounting for length:

\frac{a\cdot b}{\|a\|\|b\|}

That is why it is often used to compare embedding directions.

Enter 1.

Both vectors have shape two, but the coordinate meanings do not match:

[rooms, distance_km]
[age_years, height_cm]

Adding them would mix unrelated quantities. Shape is necessary for vector addition, but meaningful addition also needs matching coordinate roles.

Subtract the projected vector from the original vector:

[6,4] - [6,0] = [0,4]

The projection keeps the part along $[1,0]$ . The residual is the part left over.