## Euclidean space geometry: scalar product, norm and distance

Learning this material has spillover effects for Stats because everything in this section has analogs for means, variances and covariances.

### Scalar product

**Definition 1**. The **scalar product** of two vectors is defined by . The motivation has been provided earlier.

**Remark**. If matrix notation is of essence and are written as column vectors, we have The first notation is better when we want to emphasize symmetry

**Linearity**. The scalar product is linear in the first argument when the second argument is fixed: for any vectors and numbers one has

(1)

**Proof**.

**Special cases**. 1) **Homogeneity**: by setting we get 2) **Additivity**: by setting we get

**Exercise 1**. Formulate and prove the corresponding properties of the scalar product with respect to the second argument.

**Definition 2**. The vectors are called **orthogonal** if

**Exercise 2**. 1) The zero vector is orthogonal to any other vector. 2) If are orthogonal, then any vectors proportional to them are also orthogonal. 3) The **unit vectors** in are defined by (the unit is in the th place, all other components are zeros), Check that they are pairwise orthogonal.

### Norm

**Exercise 3**. On the plane find the distance between a point and the origin.

Once I introduce the notation on a graph (Figure 1), everybody easily finds the distance to be using the Pythagoras theorem. Equally easily, almost everybody fails to connect this simple fact with the ensuing generalizations.

**Definition 3**. The **norm** in is defined by It is interpreted as the *distance* from point to the origin and also the *length* of the vector .

**Exercise 4**. 1) Can the norm be negative? We know that, in general, there are two square roots of a positive number: one is positive and the other is negative. The positive one is called an **arithmetic square root**. Here we are using the arithmetic square root.

2) Using the norm can you define the distance between points

3) The relationship between the norm and scalar product:

(2)

True or wrong?

4) Later on we'll prove that Explain why this is called a **triangle inequality**. For this, you need to recall the parallelogram rule.

5) How much is If what can you say about

**Norm of a linear combination**. For any vectors and numbers one has

(3)

From (2) we have

(using linearity in the first argument)

(using linearity in the second argument)

(applying symmetry of the scalar product and (2))

**Pythagoras theorem**. If are orthogonal, then

This is immediate from (3).

**Norm homogeneity**. Review the definition of the absolute value and the equation . The norm is **homogeneous of degree 1**:

.

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