The magnitude (or length) of the vector $\mathbf{a} = \langle 4, 5, 6 \rangle$ can be found using the formula for the magnitude of a vector in three-dimensional space:

$||\mathbf{a}|| = \sqrt{a_1^2 + a_2^2 + a_3^2}$

Substituting the components of the vector $\mathbf{a} = \langle 4, 5, 6 \rangle$:

$||\mathbf{a}|| = \sqrt{4^2 + 5^2 + 6^2}$

Calculating each term:

$||\mathbf{a}|| = \sqrt{16 + 25 + 36} = \sqrt{77}$

So, the magnitude $||\mathbf{a}||$ is $\sqrt{77}$.

Would you like more details or have any questions? Here are some related questions you might consider:

1. How do you find the magnitude of a vector in two-dimensional space?
2. How would you compute the magnitude of a vector given its initial and terminal points?
3. What is the unit vector corresponding to $\mathbf{a} = \langle 4, 5, 6 \rangle$?
4. How does the magnitude of a vector relate to its components?
5. What is the geometric significance of the magnitude of a vector?

Tip: When calculating the magnitude of a vector, always square each component individually, sum them, and then take the square root of the total.