3 dimensional vector problem .

[ngkamsengpeter]

Homework Statement

Given a vector R=Rx + Ry + Rz , prove that the cos a = Rz/R where a is the angle between vector Rz and R .

Homework Equations

I just don't know how to start .

The Attempt at a Solution

 

[Dick]

Your notation isn't totally clear, but I'll assume you mean Rx, Ry and Rz to be perpendicular to each other. Then draw a triangle with sides R, Rz and Rx+Ry. Since Rz and Rx+Ry are perpendicular to each other this is a right triangle. And R is the hypotenuse. Now use trig.

 

[HallsofIvy]

Another way is to use the dot product, if you have had that in class. One definition of \vec{u}\cdot\vec{v} is |\vec{u}||\vec{v}|cos(\theta) where \theta is the angle between \vec{u} and \vec{v}. Another, equivalent, definition is that the dot prouct of u_x\vec{i}+ u_y\vec{j}+ u_z\vec{k} and v_x\vec{i}+ v_y\vec{j}+ v_z\vec{k} is u_xv_x+ u_yv_y+ u_zv_z. comparing those should give you then angle between R_x\vec{i}+ R_y\vec{j}+ R_z\vec{k} and R_x\vec{i}.