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

1 Answer
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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