Unit vector and direction cosines?

[Boris Lagutin]

I understand that we find the direction cosines (see the drawing) by dividing the vector components (i, j and k) by the vector magnitude. Ok. I can understand cosine formed by the vector magnitude and k-axis but I don't understand cosine formed by i and j - components of the vector. If I understand right cos(alpha) is formed by i-component divided by the vector PROJECTION (not the vector magnitude) on the plane devoted by i-axis and j-axis, therefore cos(beta) is formed by j-component divided by the vector projection (not the vector magnitude). So why is the vector magnitude used to define cos(alpha) and cos(beta)?

[joe]

Gday Boris,

you have a diagram a bit like the one in
http://www.animations.physics.unsw.edu. ... ors.htm#3d
You might want to consider the vector that I've called h – the projection onto the x,y plane.
Take the components here, and then consider the z components

Best
Joe

[Boris Lagutin]

Thanks a lot, Joe.

My problem was that I didn't know that there is not only "scalar projection" but and "vector projection". They are different: "scalar projection" is just a number and "vector projection" is a vector.

Have a nice weekend