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 $R_x\vec{i}$.