Direction cosine

In analytic geometry, the direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three positive coordinate axes. Equivalently, they are the contributions of each component of the basis to a unit vector in that direction.

Three-dimensional Cartesian coordinates

Vector Template:Math in Template:Tmath

Direction cosines and direction angles for the unit vector $\frac{𝐯}{|𝐯|}$

If Template:Math is a Euclidean vector in three-dimensional Euclidean space, Template:Tmath

$$𝐯=v_x{𝐞}_x+v_y{𝐞}_y+v_z{𝐞}_z,$$

where Template:Math are the standard basis in Cartesian notation, then the direction cosines are

$$\begin{array}{cccc}\alpha & =\cos a=\frac{𝐯\cdot {𝐞}_x}{\Vert 𝐯\Vert }& & =\frac{v_x}{\sqrt{v_x^2+v_y^2+v_z^2}},\\ \beta & =\cos b=\frac{𝐯\cdot {𝐞}_y}{\Vert 𝐯\Vert }& & =\frac{v_y}{\sqrt{v_x^2+v_y^2+v_z^2}},\\ \gamma & =\cos c=\frac{𝐯\cdot {𝐞}_z}{\Vert 𝐯\Vert }& & =\frac{v_z}{\sqrt{v_x^2+v_y^2+v_z^2}}.\end{array}$$

It follows that by squaring each equation and adding the results

$${\cos }^{2}a+{\cos }^{2}b+{\cos }^{2}c={\alpha }^2+{\beta }^2+{\gamma }^2=1.$$

Here Template:Math are the direction cosines and the Cartesian coordinates of the unit vector $\frac{𝐯}{|𝐯|},$ and Template:Math are the direction angles of the vector Template:Math.

The direction angles Template:Math are acute or obtuse angles, i.e., Template:Math, Template:Math and Template:Math, and they denote the angles formed between Template:Math and the unit basis vectors Template:Math.

General meaning

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 orthonormal basis vectors in terms of another set, or for expressing a known vector in a different basis. Simply put, direction cosines provide an easy method of representing the direction of a vector in a Cartesian coordinate system.

Applications

Determining angles between two vectors

Let Template:Math and Template:Math have direction cosines Template:Math and Template:Math, respectively, having an angle Template:Mvar between them. Their unit vectors are$$\begin{array}{cc}\widehat{𝐮}& =\frac{u_x}{\Vert 𝐮\Vert }{𝐞}_x+\frac{u_y}{\Vert 𝐮\Vert }{𝐞}_y+\frac{u_z}{\Vert 𝐮\Vert }{𝐞}_z={\alpha }_u{𝐞}_x+{\beta }_u{𝐞}_y+{\gamma }_u{𝐞}_z\\ \widehat{𝐯}& =\frac{v_x}{\Vert 𝐯\Vert }{𝐞}_x+\frac{v_y}{\Vert 𝐯\Vert }{𝐞}_y+\frac{v_z}{\Vert 𝐯\Vert }{𝐞}_z={\alpha }_v{𝐞}_x+{\beta }_v{𝐞}_y+{\gamma }_v{𝐞}_z\end{array}$$respectively.

Taking the scalar product of these two unit vectors yield,$$\widehat{𝐮}\mathbf{\cdot }\widehat{𝐯}={\alpha }_u{\alpha }_v+{\beta }_u{\beta }_v+{\gamma }_u{\gamma }_v.$$The geometric interpretation of the scalar product of these two unit vectors is equivalent to the projection of one vector onto another; linking the two definitions we find the following.

${\alpha }_u{\alpha }_v+{\beta }_u{\beta }_v+{\gamma }_u{\gamma }_v=\cos \theta$

There exist two choices for Template:Mvar (because cosine is even); one is acute, another is the obtuse angle between them. The convention is to choose the acute, so we take the absolute value of the scalar product.$$\theta =\arccos \left(|{\alpha }_u{\alpha }_v+{\beta }_u{\beta }_v+{\gamma }_u{\gamma }_v|\right).$$

See also

• Cartesian tensor
• Euler angles

References

• Template:Cite book
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• Template:MathWorld

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