# Question #b4b92

Feb 28, 2016

Three angles

#### Explanation:

We live in three dimensional space. In a Cartesian coordinate system we need three coordinates to uniquely define a point in space, its $\left(x , y , z\right) .$

Vector is defined as a quantity which has both magnitude and direction.

If $\vec{u}$ is a vector quantity, we need three angles which it makes with the three axes, as shown in the picture above, to uniquely define it.

If $\alpha$ is the angle between $\vec{u}$ and the $x$-axis,
$\beta$ is the angle between $\vec{u}$ and the $y$-axis and
$\gamma$ is the angle between $\vec{u}$ and the $z$-axis

Then unit vector $\hat{u}$ can be written as

$\hat{u} = \cos \alpha \hat{i} + \cos \beta \hat{j} + \cos \gamma \hat{k}$

Where $\hat{i} , \hat{j} \mathmr{and} \hat{k}$ are unit vectors in $x , y , z$ directions respectively.
And these three cosines are called the direction cosines.