Relationship between vector's length and scalar multiplication

Question

The question is:

Find vector's $(1, 1)$ length, if scalar multiplication $(v, u)$ is

1. $(v, u) = ((x_1, y_1),(x_2, y_2)) = x_1\cdot x_2 + y_1\cdot y_2$
2. $(v, u) = ((x_1, y_1),(x_2, y_2)) = (x_1 + y_1)\cdot(x_2 + y_2)$

How vector's length even related to the scalar multiplication? Thanks

Answer 1

By "scalar multiplication" this question means "dot product," as in "the multiplication of two vectors which produces the scalar". You then can use the formula $\|u\| = \sqrt{(u,u)}$ to get the length of a vector $u$

Answer 2

Every inner product (also known as scalar multiplication) induces a so called norm which expresses the 'length' of a vector, with respect to the given inner product.

Specifically, the norm is defined by $\ \|v\|^2=\langle v,v\rangle$.

Equipping a vector space with an inner product basically amounts to introducing a notion of length and a notion of enclosed angles for the vectors.