How to find the resultant of 3d vector?

[asanka000]

i have a vector ai+bj+ck

is the resultant equal to sqrt(a^2+b^2+c^2). if so how do i explain it mathematically?

i know the resultant of a 2d vector is equal to sqrt(a^2+b^2) and it can be proved from pythagoras theorem? any ideas??

 

[Petar Mali]

That is not resultant that is intensity. Yes if you have vector $$\vec{A}$$ then

$$|\vec{A}|=\sqrt{\vec{A}\cdot\vec{A}}$$

 

[Mandeep Deka]

first of all, when u say u have a vector ai+bj, sqrt(a^2+b^2) gives u the magnitude of the vector, u shouldn't call it "resultant" as such,
as far as a 3d vector is concerned u can find its magnitude using Pythagoras theorem only..

Imagine the vector (a line with the arrow mark) ai+bj+ck, in a xyz space. Then imagine u have drawn a plane consisting the vector and perpendicular to the xy plane, then u can think of a rt triangle OAB being formed with co-ordinates, O(0,0,0); A(a,b,0); and B(a,b,c). The height of the rt triangle is 'b', and its base length is nothing but sqrt(a^2+b^2),[by distance formula]. therefore the length of the hypotenuse of the triangle OAB is sqrt(a^2+b^2+^c).

this is the magnitude of ur vector ai+bj+ck!

i hope u got it!