Question

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

Answer 1

`2sqrt(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((-4)^2+8^2+6^2)`. Which simplifies down to `sqrt(116)` and then to `2sqrt(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)`.