# What do you mean by direction cosine

1. Mar 6, 2006

### denian

what do you mean by "direction cosine"

tqvm.

2. Mar 6, 2006

### arildno

If you project a unit vector down on your chosen set of axes, then the direction cosines are the length of the projection along each respective axis.

3. Mar 6, 2006

### HallsofIvy

If a vector, v, in 2 dimensions, makes angle $\theta_x$ with the x-axis, then cos($\theta_x$)i+ sin($\theta_x$)j is a unit vector in the same direction as v. If $\theta_y$ is the angle that vector makes with the y axis, then $\theta_y= \frac{\pi}{2}- \theta_x$) so sin($\theta_x$)= cos($\theta_y$ and that unit vector is cos($\theta_x$)i+ cos($\theta_y$)j.

Similarly, if a vector, v, in 3 dimensions, make angle $\theta_x$ with the x-axis, angle $\theta_y$ with the y axis, and angle $\theta_z$ with the z-axis then a unit vector in the direction of v is cos($\theta_x$)i+ cos($\theta_y$)j+ cos($\theta_z$)k.

Those are the "direction cosines" of the vector or line in the direction of the vector. Equivalently, if v is a unit vector in a given direction, its components are the direction cosines for that direction- its components are the cosine of the angle it makes with the corresponding axis.