## 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.

Figure 1. Pythagoras theorem

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)

**Proof**. 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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