Euclidean norm (length) of the vector [4 -2 -6]T is A. \[\sqrt {48} \] B. \[\sqrt {56} \] C. \[\sqrt {24} \] D. \[\sqrt {12} \]

” option2=”\[\sqrt {56} \]” option3=”\[\sqrt {24} \]” option4=”\[\sqrt {12} \]” correct=”option4″]

The correct answer is $\boxed{\sqrt{24}}$.

The Euclidean norm (also called the 2-norm, L2 norm, or Euclidean distance) of a vector $v$ is defined as:

$$\|v\|2 = \sqrt{v^Tv} = \sqrt{\sum{i=1}^n v_i^2}$$

where $v_i$ is the $i$th element of $v$ and $n$ is the number of elements in $v$.

In this case, $v = [4 -2 -6]^T$, so:

$$\|v\|_2 = \sqrt{4^2 + (-2)^2 + (-6)^2} = \sqrt{40} = \sqrt{2^3\cdot5} = 2\sqrt{2}\sqrt{5} = \sqrt{24}$$

Option A is incorrect because $\sqrt{48} = 2\sqrt{24}$. Option B is incorrect because $\sqrt{56} = 2\sqrt{28}$. Option C is incorrect because $\sqrt{24} \neq \sqrt{12}$.