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

1 Answer
LewisCroney
May 17, 2016

#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)#.

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