The problem asks us to calculate the Euclidean norm of vector $v$ for four different cases: a) $v = (4, -3)$ b) $v = (1, -1, 3)$ c) $v = (2, 0, 3, -1)$ d) $v = (-1, 1, 1, 3, 6)$

1. Problem Description

The problem asks us to calculate the Euclidean norm of vector $v$ for four different cases:

a) $v = (4, -3)$

b) $v = (1, -1, 3)$

c) $v = (2, 0, 3, -1)$

d) $v = (-1, 1, 1, 3, 6)$

2. Solution Steps

The Euclidean norm of a vector $v = (v_1, v_2, ..., v_n)$ is defined as:

$||v|| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}$

a) $v = (4, -3)$

$||v|| = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

b) $v = (1, -1, 3)$

$||v|| = \sqrt{1^2 + (-1)^2 + 3^2} = \sqrt{1 + 1 + 9} = \sqrt{11}$

c) $v = (2, 0, 3, -1)$

$||v|| = \sqrt{2^2 + 0^2 + 3^2 + (-1)^2} = \sqrt{4 + 0 + 9 + 1} = \sqrt{14}$

d) $v = (-1, 1, 1, 3, 6)$

$||v|| = \sqrt{(-1)^2 + 1^2 + 1^2 + 3^2 + 6^2} = \sqrt{1 + 1 + 1 + 9 + 36} = \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}$

3. Final Answer

a) $||v|| = 5$

b) $||v|| = \sqrt{11}$

c) $||v|| = \sqrt{14}$

d) $||v|| = 4\sqrt{3}$