# What is  || < -4 , 8 , 6 > || ?

May 17, 2016

$2 \sqrt{29}$

#### Explanation:

This operation is known as the magnitude. It represents how 'long' the vector is. You can imagine this vector starts from the origin and goes -4 in the x, 8 in the y and 6 in the z direction. Therefore, the length of this vector is $\sqrt{{\left(- 4\right)}^{2} + {8}^{2} + {6}^{2}}$. Which simplifies down to $\sqrt{116}$ and then to $2 \sqrt{29}$

This might sound confusing, but take a cuboid of width, length and height: x, y and z (respectively). It is trivial to prove that using Pythagoras that the length from the bottom left hand corner to the top right hand corner is $\sqrt{{x}^{2} + {y}^{2} + {z}^{2}}$. And now with that knowledge, that line represents this vector within that space. And so that's the magnitude.

Continue reading for a full proof of the above formula using the cuboid idea:

(Diagram at bottom):
Consider that ${x}^{2} + {y}^{2} = {a}^{2}$, by Pythagoras.
And ${a}^{2} + {z}^{2} = {d}^{2}$.
So because of our first rule, ${x}^{2} + {y}^{2} + {z}^{2} = {d}^{2}$
So finally, $d = \sqrt{{x}^{2} + {y}^{2} + {z}^{2}}$. 