Mathguru – Help – Find the direction cosines if direction ratios are given.

11.1 Direction Cosines and Ratios of a Line > No. 3

Find the direction cosines if direction ratios are given.

Post to:

Explanation:

Direction cosine

In analytic geometry, the direction cosines of a vector are the cosines of the angles between the vector and the three coordinate axes.

If v is a vector

$Description: {\mathbf v}= v_1 \boldsymbol{\hat{x}} + v_2 \boldsymbol{\hat{y}} + v_3 \boldsymbol{\hat{z}}$

where $Description: \boldsymbol{\hat{x}}, \boldsymbol{\hat{y}}, \boldsymbol{\hat{z}}$ is a basis. Then the direction cosines are

$Description: \begin{align} \alpha & = \cos a = \frac{{\mathbf v} \cdot \boldsymbol{\hat{x}} }{ \left \Vert {\mathbf v} \right \Vert } & = \frac{v_1}{\sqrt{v_1^2 + v_2^2 + v_3^2}} ,\\ \beta & = \cos b = \frac{{\mathbf v} \cdot \boldsymbol{\hat{y}} }{ \left \Vert {\mathbf v} \right \Vert } & = \frac{v_2}{\sqrt{v_1^2 + v_2^2 + v_3^2}} ,\\ \gamma &= \cos c = \frac{{\mathbf v} \cdot \boldsymbol{\hat{z}} }{ \left \Vert {\mathbf v} \right \Vert } & = \frac{v_3}{\sqrt{v_1^2 + v_2^2 + v_3^2}}. \end{align}$

(Our solved example in mathguru.com uses this concept)

Note that

α² + β² + γ² = 1

and

(α, β, γ) is the Cartesian coordinates of the unit vector $Description: \boldsymbol{\hat{v}}$

More generally, direction cosine refers to the cosine of the angle between any two vectors. They are useful for forming direction cosine matrices that express one set of orthonormalbasis vectors in terms of another set, or for expressing a known vector in a different basis.

http://en.wikipedia.org/wiki/Direction_cosine

The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.

(Our solved example in mathguru.com uses the below concept. This is our own explanation, it is not taken from Wikipedia.)

Direction ratios

For any vector r = a + b+ c its direction ratios are a : b : c. Its direction cosines are

l = a / (a² +b² +c²)

m = a / (a² +b² +c²)

n = a / (a² +b² +c²)